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Seesaw and seesaw-like scenarios : Somephenomenological implications

By

Avinanda ChaudhuriPHYS08201004006

Harish-Chandra Research Institute, Allahabad

A thesis submitted to the

Board of Studies in Physical Sciences

In partial fulfillment of requirements

For the degree of

DOCTOR OF PHILOSOPHY

of

HOMI BHABHA NATIONAL INSTITUTE

June, 2016

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for anadvanced degree at Homi Bhabha National Institute (HBNI) and is deposited in theLibrary to be made available to borrowers under rules of the HBNI.

Brief quotations from this dissertation are allowable without special permission, pro-vided that accurate acknowledgement of source is made. Requests for permission forextended quotation from or reproduction of this manuscript in whole or in part may begranted by the Competent Authority of HBNI when in his or her judgment the proposeduse of the material is in the interests of scholarship. In all other instances, however, per-mission must be obtained from the author.

Avinanda Chaudhuri

DECLARATION

I, hereby declare that the investigation presented in the thesis has been carried out byme. The work is original and has not been submitted earlier as a whole or in part for adegree/diploma at this or any other Institution/University.

Avinanda Chaudhuri

List of Publications arising from the thesis

Journal

1. A. Chaudhuri, W. Grimus and B. Mukhopadhyaya, "Doubly charged scalar decaysin a type II seesaw scenario with two Higgs triplets", JHEP 1402 (2014) 060.

2. A. Chaudhuri, N. Khan, B. Mukhopadhyaya and S. Rakshit, "Dark matter candidatein an extended type III seesaw scenario", Phys. Rev. D 91 (2015) 055024.

3. A. Chaudhuri and B. Mukhopadhyaya, "CP -violating phase in a two Higgs tripletscenario: Some phenomenological implications" , Phys. Rev. D 93 (2016) no.9, 093003.

Conferences

"Texture zeros and type II seesaw mechanism",A. Chaudhuri,Nu Horizon, Harish-Chandra Research Institute, Allahabad, 17-19 th March, 2016.

Avinanda Chaudhuri

Dedicated to

my parents

Acknowledgements

It is not possible for me to describe in words the inspiration, encouragement and love Ireceived from my parents, without which it would have been impossible for me to reachupto the stage where I am now.

I would like to express my deepest gratitude to my thesis supervisor ProfessorBiswarup Mukhopadhyaya for his invaluable guidance at every step of my researchwork. He has encouraged me to think and work independently all along.

I have learnt a lot from my collaborators. I would like to thank Prof. Walter Grimus forhis wonderful guidance during the period of our collaboration. I take this opportunity tothank Prof. Subhendu Rakshit and Mr. Najimuddin Khan for the collaborating work. Iwould like to acknowledge the financial support I have received from the Regional Centrefor Accelerator-based Particle Physics (RECAPP), HRI and Department of Atomic Energy,Govt. of India.

I thank all my friends at HRI, seniors, batchmates, juniors and post doctoral fellows fortheir support. I would like to specially thank Dr. Tanumoy Mondal and Dr. SubhadeepMondal for their understanding and numerous help.

I would like to express my sincere regards to my teacher Prof. Asoke Sen, under whoseguidance I have done a project work during my course of study at HRI, for his inspiration.

My regards are due to Dr. Aditi Mukhopadhyaya for her help in variours ways duringmy stay at HRI, Allahabad.

I want to take this opportunity to express my deepest respect to Prof. Sourov Roy (orSourov da, as I call him) for being a constant source of inspiration to me.

I would like to convey my regards to my teachers at HRI, Prof. Rajesh Gopakumar,Prof. Sumathi Rao, Prof. Raj Gandhi, Prof. Asesh Krishna Datta, Prof. Ujjwal Sen, Prof.Sandhya Choubey, Prof. Aditi Sen(De), Prof. L. Sriramkumar, Prof. Satchidananda Naik,Prof. Dileep Jatkar, Prof. Pinaki Majumdar, Prof. V. Ravindran, Prof. Tirthankar RoyChoudhury, for their help during my stay at HRI.

My thanks are also due to Rukmini di (Dr. Rukmini Dey) for the music evenings thatI enjoyed in her house.

Lastly, I would like to thank all the employees of HRI, Allahabad for their help duringmy research work.

Contents

Synopsis 5

LIST OF FIGURES 11

LIST OF TABLES 13

1 Introduction 15

1.1 Standard Model and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 Particle content of SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Origin of mass : Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Need for new physics beyond SM . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Neutrino Physics 23

2.1 Neutrino masses and mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Experiments with neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Motivations for physics beyond the SM . . . . . . . . . . . . . . . . . . . . . 30

2.4 Models of neutrino mass generation . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Texture zeros of neutrino mass matrices . . . . . . . . . . . . . . . . . . . . . 35

3 Seesaw mechanism 39

3.1 Introduction to seesaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Type I seesaw : Fermion singlets . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Type II seesaw : Scalar triplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Type III seesaw : Fermion triplets . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5 Two-zero texture and the inadequacy of a single triplet . . . . . . . . . . . . 46

1

CONTENTS

4 Doubly charged scalar decays in a type II seesaw scenario with two Higgstriplets 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 The scenario with a single triplet . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 A two Higgs triplet scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Benchmark points and doubly-charged scalar decays . . . . . . . . . . . . . 57

4.5 Usefulness of H++1 −→ H+

2 W+ at the LHC . . . . . . . . . . . . . . . . . . . . 59

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 CP-violating phase in a two Higgs triplet scenario : some phenomenologicalimplications 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 A single scalar triplet with a CP-violating phase . . . . . . . . . . . . . . . . 69

5.3 Two scalar triplets and a CP-violating phase . . . . . . . . . . . . . . . . . . 71

5.4 Benchmark points and numerical predictions . . . . . . . . . . . . . . . . . . 76

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Dark matter 91

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 Evidences for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.3 Some properties of dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.4 Candidates for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5 Calculation of dark matter relic density . . . . . . . . . . . . . . . . . . . . . 97

6.5.1 Standard calculation of relic density . . . . . . . . . . . . . . . . . . . 98

6.5.2 Coannihilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.6 Search for dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.6.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.6.2 Indirect search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.6.3 Accelerator search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 A dark matter candidate in an extended type III seesaw scenario 107

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

2

CONTENTS

8 Summary and conclusions 119

References 121

3

SYNOPSIS

Introduction

The Standard Model (SM) of particle physics has been extremely successful in describingmost of the observed phenomena in particle physics. However, there is a strong moti-vation for physics beyond the SM, to address some yet unexplained observations, whichinclude, small masses and mixing pattern of neutrinos, existence of Dark Matter (DM)and Dark energy in the Universe, matter-antimatter asymmetry of Universe and so on.Also, in the Large Hadron Collider (LHC) era, we hope to test the theory beyond the SM,which describes physics at the TeV scale. So, physics beyond the SM is a serious studyfor any aspiring physicist.

The electroweak symmetry SU(2)L×U(1)Y is broken by the vacuum expectation value(vev) of the Higgs doublet v ' 246 GeV, which gives mass to the gauge bosons and all thefermions except the neutrino. There is no SU(2)L singlet neutrino-like field in the SM, thatcan generate a Dirac mass term for the neutrino. Thus, neutrino is massless in the SM, attree level. It can be seen that, to all orders in perturbation theory within SM, the neutrinois massless and also their masses remain zero even in the presence of non-perturbativeeffects.

Thus, one must seek physics beyond the SM, to explain observed evidence for neu-trino masses. While there are many possibilities that lead to small neutrino masses, in thisproposed thesis we focus on seesaw frameworks — which are considered as the most el-egant way of explaining the smallness of neutrino mass. There are only four mechanismsin physics beyond the SM, to generate Majorana neutrino masses at tree level. They arecalled the — Type I, II, III seesaw mechanism and the Inverse Seesaw mechanism. Type Iseesaw mechanism is the simplest possible extension of SM that leads to non-zero massfor the neutrino. In this case, only a right-handed neutrino is added to the SM. Sub-eVneutrino mass can be obtained as functions of Dirac Yukawa couplings, the SM Higgsvev and heavy mass of the right-handed neutrino. Type II seesaw is a way to generatenon-zero neutrino masses without using the right-handed neutrino. In this framework,an SU(2)L scalar triplet field (∆++,∆+,∆0) , with Y = 2 hypercharge, is included in the

SYNOPSIS

SM. This allows an additional ∆L = 2 Yukawa coupling between the scalar triplet and thelepton doublet. When the neutral member of the scalar triplet ∆0 acquires a vev, then thisyukawa term produces small neutrino masses in terms of combinations between yukawacoupling constants and triplet vev. In the case of Type III seesaw mechanism, at least twoextra matter fields (normally taken as fermion triplets) with zero hypercharge are addedto the SM. This in turn produces sub-eV neutrino mass as function of Yukawa coupling,vev of SM Higgs and heavy mass of the fermion triplets. Another realization of seesawmechanism is the so-called ’Inverse Seesaw mechanism’, where small neutrino mass ariseas a result of new physics at TeV scale, which may be probed at the LHC experiments. Theimplimentation of this mechanism requires the addition of three right-handed neutrinosand three extra singlet neutral fermions to the three active neutrinos in SM. Then, consid-ering a definite hierarchy between various mass terms of the right-handed neutrinos andsinglet states, small neutrino mass can be generated.

Of all these scenarios, the works presented in this proposed thesis are based on Type IIand Type III seesaw framework. The content of the proposed thesis is divided into twoparts.

• In the first part of the thesis, phenomenology of doubly-charged scalar decays inType II seesaw framework and usefulness of these decays at the LHC has been de-scribed.

• The second part of the thesis describes the prediction of a Dark Matter (DM) candi-date within a scenario inspired by the Type III seesaw framework.

The following sections contain brief descriptions of the above topics.

Doubly-charged scalar decays in Type II seesaw framework

The motivation behind these works is two-fold. Firstly, strong evidence has accumulatedin favour of neutrino oscillation from the solar, atmospheric, reactor and accelerator neu-trino experiments over the last few years. A lot, however, is yet to be known, includingthe mass generation mechanism and the absolute values of the masses, as opposed tomass-squared differences which affect oscillation rates. A gateway to information of bothof the above kinds is some idea about the light neutrino mass matrix, in a basis where thecharged lepton mass matrix is diagonal. Here, too, in absence of very clear guidelines,various ’textures’ for the neutrino mass matrix are often investigated. A possibility that

6

SYNOPSIS

frequently enters into such investigations is one where the mass matrix has some zeroentries. Such ’zero textures’ lead to a higher degree of predictability and inter-relationbetween mass eigenvalues and mixing angles, by virtue of fewer free paramaters. In thecontext of Majorana neutrinos which have a symmetric mass matrix, various texture ze-ros have thus been studied from a number of angles. Of them, two-zero textures haverather wide acceptability from the viewpoint of explaining observed data. It has beenshown that, none of the seven possible two-zero-texture cases can be achieved by assum-ing only one scalar triplet in the theoretical framework of Type II seesaw. It has also beenshown that, the contradictions that appear with a single triplet can be avoided, if twoor more scalar triplets are present in the theory. So, we see that, there is need for an ex-tended scalar triplet sector, if Type II seesaw has to be consistent with two-zero-textures ofneutrino mass matrix. Secondly, when a single triplet is present in a Type II seesaw frame-work, then most conspicuous signal consists in the decay of doubly-charged scalar into apair of same-sign leptons and alternatively, the decay into a pair of same-sign W-bosons,which is dominant in a complementary region of the parameter space. It occured to usthat, a third decay channel is also possible, where, a doubly-charged scalar decays intoa singly-charged scalar plus a W boson of same sign. But, such a decay mode is usuallysuppressed, since the underlying SU(2) invariance implies relatively small mass splittingamong the members of a triplet. However, when more than one triplet of a similar natureare present, and mixing among them is allowed, a transition such as the third decay chan-nel mentioned above is possible between two scalar mass eigenstates. So, motivated bythe necessity of more than one triplet in applying the Type II seesaw to a class of neutrinomass textures and to examine whether the SU(2) gauge coupling driven decay where adoubly-charged scalar decays into a singly-charged scalar plus a W boson of same signreally overrides the other two decay channels as mentioned above, we take up the case oftwo coexisting triplets in our work.

We have shown that, this decay really dominates over all other decays, by choosingseveral benchmark points(BPs) over a wide range of parameter space and by taking allthe vevs and coefficients of Lagrangian to be real. In other words, in our first attempt,we have neglected all the CP-violating effects. We have also pointed out the distinctionbetween having a single triplet and two triplets in the scenario, at the LHC, by defin-ing ratios of events emerging from two, three and four-lepton final states with missingtransverse energy(MET). We found that, after applying suitable cuts to suppress the SMbackground, these ratios are widely different in two-triplet case as compared to single-triplet model — this provides a way to distinguish between the two scenarios at the LHC.

7

SYNOPSIS

To see the phenomenology including CP-violation, we make a minimal extension ofthe simplified scenario by postulating one CP-violating phase to exist. This entails a com-plex vev for any one triplet, at the same time, there is a complex phase in the coefficient ofthe Lagrangian. We found that phases of complex vev and coefficients are inter-related.We have chosen our BPs over a wide range of parameter space and adjusted the freeparameters in such a way, that a new decay channel of doubly-charged scalar openedup, in addition to the previously mentioned one, as a result of increase in mass sepa-ration between heavier and lighter doubly-charged scalars. We observed that the heav-ier doubly-charged scalar decays dominantly into lighter doubly-charged scalar plus SMHiggs. This, in turn, opens up a spectacular signal at LHC, especially when the lighterdoubly-charged scalar mostly decays into two same sign leptons, leading to the decayof heavier doubly-charged scalar into a pair of same-sign leptons plus SM Higgs. Wenoticed that, when the phase space needed for this decay is not available, the processwhere the heavier doubly-charged scalar decays into a singly-charged scalar plus a same-sign W boson, dominates over all other remaining decays — this situation has alreadybeen discussed. We pointed out some rather interesting effect of presence of phase. If themass differences do not allow the above mentioned decays when the phase is absent, theyeventually open-up with the increase in phase of triplet vev, when all other parametersare at fixed values. Another notable effect of presence of phase is that, the contribution oftriplet vev to neutrino masses gets suppressed. This results in an increase in correspond-ing Yukawa couplings. This is reflected in the fact that, the decay of a doubly-chargedscalar into two same-sign leptons too has non-negligible branching ratios for some BPs.We have simulated same-sign di-lepton final states from both the doubly-charged scalarsto confirm our findings. Also, we have ensured that the lightest neutral scalar must havemass ∼ 125 GeV for all BPs.

In short, in these works we have attempted to explore the trademark signals of scalartriplets in the Type II seesaw framework, that are contained in the doubly-charged com-ponents.

Prediction of a new Dark Matter (DM) candidate

We have already emphasized the importance of Type III seesaw mechanism in the contextof generating small mass of neutrino. On the other hand, there is strong empirical evi-dence for the existence of dark matter in the Universe. A simple and attractive candidatefor DM should be a new elementary particle that is electrically neutral and stable on cos-

8

SYNOPSIS

mological scales. Our proposal is an extension of the type-III seesaw model for neutrinomasses. As two fermion triplets are required to generate neutrino masses, we need to in-troduce a third triplet to take care of the DM candidate. This triplet needs to be protectedby a Z2 symmetry to ensure stability of the DM particle. This third triplet does not con-tribute to neutrino mass generation through seesaw, because, it is odd under imposed Z2

symmetry. The SU(2) symmetry will in general demand the charged and neutral compo-nents of the fermion triplet to be degenerate. However, we need a mass splitting betweenthe charged and neutral component to obtain a DM particle of mass of the order of theelectroweak scale. In some models this mass splitting between the charged and neutralcomponents of the fermion triplet have been obtained through radiative correction. Themass of the charged member of the triplet in such a case has to be well above ∼ TeV,in order to avoid fast t-channel annihilation and a consequent depletion in relic density.As a result, such models end up with somewhat inflexible prediction of DM mass in theregion 2.5 − 2.7 TeV. We propose, as an alternative, two Z2-odd fermion fields: a Y = 0

triplet and a neutral singlet that appears as a heavy sterile neutrino. All SM fields andthe first two fermion triplets are even under this Z2 symmetry. Mixing between neutralcomponent of the triplet and the sterile neutrino can yield a Z2 odd neutral fermion statethat is lighter than all other states in the Z2 odd sector. This fermionic state emerges as theDM candidate in this work. The requisite rate of annihilation is ensured by postulatingsome Z2 preserving dimension-five operators. Choice of a light sterile neutrino can leadto light DM candidates, in our model, we worked with DM masses of more than 62.5 GeVto avoid an unacceptably large invisible Higgs decay width.

The DM mass, relic density has been calucated as function of the five independentparameters present in the theory, given by — Mass of the fermion triplet (MΣ), Mass ofsterile neutrino (Mνs) and three coefficients of the dimension-five operators introducedin the theory. Two different situations may arise : (i) MΣ Mνs and (ii) MΣ ' Mνs .In the first case, self annihilation of DM mainly contributes in the calculation of relicdensity. While, in the second case, coannihilation between the charged components ofthe triplet, the neutral eigenstate of higher mass and the dark matter candidate becomesimportant in the calculation of relic density. We have shown by varying one parameterat a time and keeping other parameters at fixed values that it is possible to obtain a wideregion of DM mass from 200 GeV to TeV range that satisfies the correct relic density rangeand the constrains imposed on WIMP-nucleon cross section by XENON 100 and LUXdata. We also got a large region of parameter space that satisfies the correct relic densityrequirement and the bound imposed by XENON 100 and LUX data on WIMP-nucleon

9

SYNOPSIS

cross section in both the cases of self-annihilation and co-annihilation of DM candidate.Dark matter direct detection experimental results are rather conflicting. While CDMS,DAMA, CoGeNT, CRESST point towards a DM mass of ∼ 10 GeV, XENON 100 and LUXclaim to rule out these observations. Given this scenario, we keep our options open andexplore all DM masses allowed by experimental observations.

10

List of Figures

4.1 Variation of mass difference between the doubly and singly-chargedscalars, for various values of the parameter h. . . . . . . . . . . . . . . . . . . 53

5.1 Variation of mass difference between (left panel) H++1 and H++

2 and (rightpanel) H++

1 and H+2 with phase of triplet α for BP 1 of scenario 1 and BP 3

of scenario 2 for α = 30, BP 4 of scenario 2 for α = 60 . . . . . . . . . . . . 80

5.2 Invariant mass distribution of same sign di-leptons for (left panel) α = 60

and (right panel) α = 65 for BP 4 of Scenario 2 . . . . . . . . . . . . . . . . . 87

5.3 Invariant mass distribution of same sign di-leptons for chosen benchmarkpoints. In the top (left panel) BP 3 and (right panel) BP 4 of Scenario 2 forα = 30. In the middle (left panel) BP 3 and (right panel) BP 4 of Scenario 2for α = 45. In the bottom, BP 4 of Scenario 2 for α = 60. . . . . . . . . . . . 88

7.1 Relic density Ωh2 vs. dark matter mass Mχ keeping αΣνs fixed at 2.8 TeV−1,2.9 TeV−1 and 3.0 TeV−1. Mχ is varied by changing the remaining parameterανs . We use MΣ = 1 TeV, αΣ = 0.1 TeV−1, Mνs = 200 GeV. The blue bandcorresponds to 3σ variation in relic density according to the WMAP + Planck data. 112

7.2 Contour plot in αΣνs – ανs plane for fixed values of MΣ = 1 TeV, Mνs = 200 GeV,αΣ = 0.1 TeV−1. In (a) DM relic density is projected onto the plane, whereasin (b), we project DM mass (in GeV). In both plots the red solid line representsΩh2 = 0.1198 and the dashed red lines correspond to the 3σ variation in Ωh2.Darker region corresponds to lower values of Ωh2 or Mχ. . . . . . . . . . . . . . . 113

11

LIST OF FIGURES

7.3 (a) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ fixed at0.1 TeV−1. Mχ is varied by changing the remaining parameters. In such a sce-nario, DM annihilates via self-annihilation only. Mνs is varied between 150 and1000 GeV. (b) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ

fixed at 0.1 TeV−1. Mχ is varied by changing the remaining parameters. Herecoannihilation between the DM candidate with the charged components of thetriplet Σ±, the neutral eigenstate of higher mass provides the dominating contri-bution to relic density. In both the figures different direct detection experimentalbounds are indicated and the red points correspond to the 3σ variation in relicdensity according to the WMAP + Planck data. . . . . . . . . . . . . . . . . . . 114

7.4 (a) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ fixed at0.01 TeV−1. Mχ is varied by changing the remaining parameters. Here self-annihilation of the DM candidate provides the dominating contribution to relicdensity (b) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ

fixed at 0.01 TeV−1. Mχ is varied by changing the remaining parameters. Herecoannihilation between the DM candidate with the charged components of thetriplet Σ±, the neutral eigenstate of higher mass provides the dominating contri-bution to relic density. In both the figures different direct detection experimentalbounds are indicated and the red points correspond to the 3σ variation in relicdensity according to the WMAP + Planck data. . . . . . . . . . . . . . . . . . . 115

12

List of Tables

4.1 Charged scalar masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Neutral scalar masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.3 Decay branching ratios and production cross sections for doubly-charged

scalars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 Cuts used for determination of ratios of events r1 and r2. The subscript T

stands for ‘transverse’ and η denotes the pseudorapidity. . . . . . . . . . . . 644.5 Ratio of events r1, r2 for two-triplet and single-triplet scenario respectively

for benchmark point 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 Charged scalar masses for phase α = 30. . . . . . . . . . . . . . . . . . . . . 805.2 Neutral scalar masses for phase α = 30. . . . . . . . . . . . . . . . . . . . . 815.3 Decay branching ratios and production cross sections for doubly-charged

scalars for phase α = 30. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4 Charged scalar masses for phase α = 45. . . . . . . . . . . . . . . . . . . . . 835.5 Neutral scalar masses for phase α = 45. . . . . . . . . . . . . . . . . . . . . 835.6 Decay branching ratios and production cross sections for doubly-charged

scalars for phase α = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.7 Charged scalar masses for phase α = 60. . . . . . . . . . . . . . . . . . . . . 855.8 Neutral scalar masses for phase α = 60. . . . . . . . . . . . . . . . . . . . . 855.9 Decay branching ratios and production cross sections for doubly-charged

scalars for phase α = 60. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.10 Number of same-sign dilepton events generated at the LHC, for the bench-

mark points corresponding to Figure 3. The integrated luminosity is takento be 2500fb−1, for

√s = 13TeV . . . . . . . . . . . . . . . . . . . . . . . . . . 87

13

Chapter 1

Introduction

1.1 Standard Model and beyond

The Standard Model (SM) of particle physics encompasses our current knowledge of el-ementary particles. The SM is a successful quantum field theory, based on the SU(3)C ×SU(2)L × U(1)Y gauge group, which describes the interactions among elementary parti-cles and three of the four fundamental interactions. In Nature, all the known phenomenaobserved so far, can be described in terms of four fundamental forces :

• Strong Interaction

• Weak Interaction

• Electromagnetic interaction

• Gravitational Interaction

The SU(3)C is the gauge group of strong forces, where ′C ′ denotes the color quantumnumber carried by quarks as well as gluons, which are mediators of strong interaction.The symmetry group corresponding to electroweak interaction is SU(2)L×U(1)Y , where′L′ refers to the left-chirality of fermions and ′Y ′ stands for the weak hypercharge, whichis defined by the relation

Q = T3 +Y

2(1.1)

′Q′ and T3 being respectively the electric charge and third component of the weak isospinof the fields involved.

So, SM describes only the first three of four fundamental forces previously mentioned.Gravity effects are negligible at the highest energy scale of particle accelerator experi-

15

CHAPTER 1. INTRODUCTION

ments performed till date and therefore its exclusion does not affect the explanation ofwhatever has been observed so far in the world of fundamental particles.

1.2 Particle content of SM

The fermionic sector of the SM consists of six types each of quarks and leptons, whichcome in three generations or flavors. The transformation properties of these fields underSM gauge group are determined by their respective charges under the gauge groups. Thequark fields transform as triplets (fundamental representation) of SU(3)C , whereas theleptons are singlet under this gauge group. The leptons do not possess any color chargeand hence they do not take part in strong interactions. SM being a chiral theory, treats left-handed and right-handed fermion fields differently with respect to SU(2) × U(1) gaugeinteractions. The left-handed fields transform as doublets (fundamental representation)under SU(2), while the right-handed fields are singlets under this group. Thus we havethe particle content for the quark sector as follows

Doublets :

(u

d

)L

;

(c

s

)L

;

(t

b

)L

Singlets : uR, dR; cR, sR; tR, bR.

For the leptonic sector we have

Doublets :

(νe

e

)L

;

(νµ

µ

)L

;

(ντ

τ

)L

Singlets : eR; µR; τR.

The left-handed chiral component of the field Ψ is defined as ψL = PLψ = [(1 − γ5)/2]ψ

and for the right-handed one, ψR = PRψ = [(1 + γ5)/2]ψ. With the convention followedso far, the lepton doublets will have hypercharge (−1) and for lepton singlets it is (−2).The quark doublets are of hypercharge +1/3, for up-type quark singlets it is +4/3 anddown-type quark singlets will have hypercharge (−2/3).

The gauge sector of SM contains eight massless vector fields Gaµ(a = 1, 2, ...8), known

as the gluons, the gauge bosons of SU(3)C . The gauge bosons corresponding to the bro-ken SU(2)L × U(1)Y group are, γ(photon), W± and Z. The gluons are electrically neutraland carry color quantum numbers. So, they have self-interactions (both trilinear andquartic). The W± are massive, charged particles and Z boson is massive but electricallyneutral. The W± and Z bosons are self-interacting also, whereas the photon is non self-interacting, massless and neutral.

16

1.3. ORIGIN OF MASS : HIGGS MECHANISM

1.3 Origin of mass : Higgs mechanism

The particle spectrum of SM described so far is incomplete. Till now, we have consideredonly those interactions among the particles, which arise as a consequence of local gaugeinvariance. However, invariance under SU(2)L × U(1)Y gauge group implies that allthe fermions as well as gauge bosons have to be massless. Introduction of explicit massterms for them threatens to destroy the good features of SM, such as, renormalizabilityand unitarity. The solution to this, is to generate the mass term in the Lagrangian bybreaking the gauge symmetry spontaneously, that is to say, not in the original Lagrangianbut through a selective (non-symmetric) choice of the vacuum. This phenomena is knownas ’Spontaneous Symmetry Breaking’(SSB) [1–9].

It has been observed that in the SM the electric charge is conserved and therefore theconcerned gauge group of electromagnetism i.eU(1)em is an exact symmetry of the theory.So under SSB, we should have the following symmetry breaking pattern

SU(2)L × U(1)YSSB−−→ U(1)em (1.2)

This breaking can be achieved by introducing a spin 0 scalar field Φ that transforms asa doublet under the SU(2)L with U(1)Y hypercharge +1. This complex scalar doublet isdefined as

Φ =

(φ+

φ0

). (1.3)

The scalar potential involving the scalar doublet, which also contains self-interaction termfor the doublet, is written as

V (Φ) = µ2Φ†Φ + λ(Φ†Φ)2 . (1.4)

The self-interaction of Φ is such that gauge invariance is broken spontaneously. Thevacuum expectation value (vev) v of the neutral component of doublet is expressed interms of the mass parameter µ and the self-interaction strength λ and is given by

v =

√−µ2

λ. (1.5)

for µ2 < 0 and λ > 0.If the scalar field is shifted with respect to the above vev, then, written in terms of the

shifted field, the gauge invariance of the theory appears to be broken. The U(1)em invari-ance, however, remains unbroken, since the non-zero vev pertains only to the neutral partof the scalar field, which has no electromagnetic interactions.

17


The resulting mass spectrum consists of massive fermionic and gauge fields as well asa neutral scalar particle known as Higgs boson, which has recently been discovered at theLarge Hadron Collider (LHC), CERN, Geneva and has mass∼ 125 GeV.

Masses of the gauge bosons are obtained from the Lagrangian

LΦ = (DµΦ)†DµΦ− V (Φ), (1.6)

where Dµ is defined by

Dµ = ∂µ − ig1Y

2Bµ − ig2

σa

2W aµ − ig3

λb

2Gbµ, (1.7)

where g1, g2 and g3 are the gauge couplings for U(1)Y , SU(2)L and SU(3)c groups respec-tively. The masses of gauge bosons are obtained as

• mγ = 0

• mW = 12g2v

• mZ = 12v√g2

1 + g22

The weak mixing angle θW , also called Weinberg angle, which gives the relationshipbetween the masses of W and Z bosons, is defined as

θW ≡ tan−1

(g1

g2

). (1.8)

The massless vector fields that corresponds to photon couples with matter fields withelectromagnetic coupling constant ’e’, the electric charge. This is related to the SU(2)L

and U(1)Y couplings in the following way

g2sinθW = g1cosθW = e (1.9)

and lastly, the ρ-parameter, which measures the relative strengths of neutral and chargedcurrent interactions in the theory, is given by

ρ =m2W

m2Zcos

2θW≈ 1. (1.10)

Masses of fermions are generated via gauage invariant Yukawa interactions betweenscalar and fermionic fields :

LYuk =∑

i,j=generation

(−Y u

ij QiΦuj − Y dijQiΦdj − Y l

ijLiΦej + h.c.), (1.11)

18

1.4. NEED FOR NEW PHYSICS BEYOND SM

where Φ = iσ2Φ∗ and Q and L represents the quark and lepton doublets respectively andY u, Y d, Y l are the Yukawa coupling matrices for the up-quark, down-quark and chargedleptons respectively. After the field Φ gets the vev v, the Yukawa Lagrangian takes theform of mψψLψR with the mass matrices,

muij ∝ vY u

ij , mdij ∝ vY d

ij , mlij ∝ vY l

ij. (1.12)

These mass matrices are in flavor basis and are to be diagonalized to get the mass basis.These Yukawa couplings are free parameters in SM and are fixed by the masses of thecorresponding fermions. It should be noted that neutrinos do not have any mass termdue to the absence of their right-handed partners.

The predictions of the SM have been tested to a high degree of precision experimentscarried out at high-energy colliders like the LEP, Tevatron and currently at the LHC, aswell as in low-energy experiments of flavor physics. In almost all cases, experimentalobservations are in accordance with the predictions of SM. The long elusive Higgs bosonhas also been discovered at the LHC [10, 11]. Apart from a few discrepancies (e.g., theanomalous magnetic moment of muon) the SM is the most consistent model of particlephysics till date. This has established SM as a starting point, as far as building fundamen-tal theoretical models of Nature is concerned. In spite of the huge success of SM, however,there are some observations that encourage us to go beyond the SM. In the next sectionwe point out some of those observations.

1.4 Need for new physics beyond SM

In the following we outline some observations which indicate that the SM suffers fromsome drawbacks and is not a complete theory to describe the particle domain.

• There are many free parameters (∼ 20) in SM, which can only be fixed throughexperiments. All the masses, couplings and mixing parameters in the quark sectorare described by these free parameters of the theory and there is no explanation asto why the parameters have such values.

• The structure of fermionic sector in SM remains unexplained. The masses of the SMfermions range from sub-eV (for neutrinos) to over hundred GeV (for top quark).There is no satisfactory fundamental explanation for this huge hierarchy in massesof fermions. The fermions also come in three ’generations’, with higher generation

19


having higher mass. Such a replication is not predicted or explained by anythingwithin the SM.

The mixing in the quark sector also has a generational structure, i.e, the largestmixing occurs between the generations one and two, followed by mixing of twoand three and finally, mixing between one and three, which are the feeblest ones.SM does not explain this pattern.

• It is important to ensure that tree-level values of the various SM parameters are sta-ble. The inclusion of higher order terms in general leads to radiative corrections thatmodify the couplings and masses via the renormalization procedure. Any quadraticcorrection to the mass for gauge bosons are tamed by the gauge symmetry. Forfermions chiral symmetry does this task, leaving only a weak, logarithmic depen-dence. However, the Higgs boson being a scalar does not have any such way outto cancel quadratic corrections and its mass would be driven to the scale of newphysics. For example, correction due to top quark contribution, shifts the mass ofHiggs from its tree level value by,

∆m2H = −|Yt|

2

4π2Λ2cutoff (1.13)

where Λcutoff is some cut off scale up to which SM is well-behaved and beyondwhich some new physics comes in and Yt is the top-quark Yukawa coupling, due towhich the correction is largest. Now, if Λcutoff ∼ Mpl, where Mpl is the Planck scale(∼ 1019 GeV), then corrections to Higgs mass squared reach upto 1038GeV 2. There-fore, in order to maintain a Higgs mass consistent with experimental measurements,we need to add counter-terms to the Higgs mass squared, so that the divergencescancel out. This obviously requires a large fine-tuning of the parameters involved.This is called the Fine-tuning/ Naturalness/ Hierarchy problem.

This so called Hierarchy problem has motivated several new physics scenarios,the most popular one being the existence of Supersymmetry (SUSY), which relatesbosonic and fermionic degree of freedom.

• Unification of the electroweak and strong couplings, is the so called Grand UnifiedTheories (GUT), cannot be achieved within the SM framework. The running of thesecouplings are such that they do not exactly unify at any given energy scale, if theirevolution all the way up is controlled by SM interactions alone.

20

1.4. NEED FOR NEW PHYSICS BEYOND SM

• Gravity, one of the four fundamental forces of Nature, which becomes importantnear Planck scale, is not included in SM. SM may at best be treated as an effectivetheory up to Planck scale and new physics should appear at that scale. One ofthe alternatives is String theory, which hope to predict a quantized description ofgravity.

• The first evidence for Dark Matter (DM) came from the measurement of rotationcurves of galaxies. The rotation curves were found to fit with the hypothesis thatvisible part of the galaxy was immersed in a halo of invisible matter i.e Dark Matter.In fact, the Universe consists of only 4.9% of visible matter and the rest consists ofDM and Dark Energy. There is no such particle in SM which can fit in properly toexplain this DM riddle.

• SM can not explain the observed baryon asymmetry of the Universe, namely, whywe have more matter than anti-matter. This will perhaps require a level of CP-violation, which is not present in the SM.

• Lastly, one of the most important findings suggesting the existence of physics be-yond the SM is the evidence for non-zero neutrino masses and mixings, observedin the form of neutrino oscillation. In SM, a neutrino is massless because of theabsence of the corresponding right-handed partner. In principle, neutrino massescan be easily accommodated in the SM framework by postulating the existence ofright-handed heavy sterile neutrinos. But, the extreme smallness of neutrino mass,which are many orders of magnitude smaller than all the fermion masses, calls fora deeper understanding. Also, the bi-large mixing pattern of neutrinos, as evidentfrom oscillation data, is very different from what is noticed in the quark sector. Thissuggest some new underlying mechanism. The question as to, why there is such alarge difference between mass of neutrino and that of the charged lepton belongingto the same SU(2)L multiplet is still a mystry. If neutrinos are found to be Majoranaparticles, then lepton number would be violated by their mass-term, which mightgive us a hint of physics beyond the SM. Several models for explaining the neutrinomass and mixing have been put forward, starting from seesaw mechanisms, largelymotivated by high-scale grand unified theories, to low-scale models like SUSY withlepton number violation. But the questions remain yet to be answered.

We have mentioned some of the shortcomings of SM. Clearly, we need some theory be-yond SM (BSM) to address these loop-holes. With this motivation we proceed to study

21


some BSM models based on seesaw mechanism, which might become helpful in under-standing and overcoming some of the incompletenesses of the SM.

22

Chapter 2

Neutrino Physics

2.1 Neutrino masses and mixing

Neutrinos are the most elusive of the known fundamental particles. They are neutral,spin-half fermions and they do not possess any color charge. As far as it is known, theyonly interact with charged fermions and massive gauge bosons through weak interac-tions. For this weakly interacting nature, neutrinos can be observed and studied experi-mentally only if there are very intense neutrino sources and availability of large detectors.

The history of neutrinos dates back to the early 1930’s, but they were first observedonly in the 1950’s in the experiments carried out by Reines and Cowan [12]. In 1962, ex-periments at Brookhaven National Laboratory made a surprising discovery that neutri-nos produced in association with muons do not behave the same way tas those producedin association with electrons. A second type of neutrino (the muon neutrino) was discov-ered in these experiments [13]. The evidence for third neutrino flavor eigenstate i.e thetau neutrino (ντ ) was obtained only in 2001, when the DONUT ("Direct Observation ofNu Tau") experiment at Fermilab [14] managed to record a handful of νe + X → ντ + Y

events. For a long time, it was believed that neutrinos are massless, making them drasti-cally different from other SM spin-half counterparts like the charged leptons (e, µ, τ ) andthe quarks (u, d, s, c, t, b), which have mass. This myth has been broken in the late 1990’s,through the strong indication of non-zero neutrino masses through oscillation phenom-ena, which opened the floodgate to many fresh questions in neutrino physics.

Various experiments with solar [15–20], atmospheric [21, 22], accelerator [23–25] andreactor [26–31] neutrinos have established beyond doubt, the phenomena of neutrino os-cillation. More explicitly, a neutrino produced in a well-defined flavor state (say, a muonneutrino νµ) has a non-zero probability of being detected in a different flavor state (say

23

CHAPTER 2. NEUTRINO PHYSICS

νe i.e electron neutrino). This probability of flavor-transition between flavor eigenstatesdepends on the neutrino energy and the distance travelled between the source and the de-tector. The simplest and only consistent explanation of almost all neutrino data collectedtill date is a phenomenon referred to as "neutrino mass-induced flavor oscillation". Theseneutrino oscillations, in turn imply that neutrinos have non-degenerate masses and neu-trino flavor eigenstates are different from neutrino mass eigenstates, that is to say, thisindicates lepton mixing.

So, if the neutrino masses are distinct and lepton mixing takes place, a neutrino fla-vor eigenstate can be produced as a superposition of neutrino mass eigenstates via weakinteraction i.e a neutrino να, with a well-defined flavor, has non-zero probability to bemeasured as a neutrino of a different flavor, say, νβ , where α, β = e, µ, τ . The oscillationprobability between the two flavor states are denoted as Pαβ and it depends on the propa-gation distance (L), neutrino energy (E) and mass-squared differences between neutrinos,(∆m2

ij ≡ m2i − m2

j ; i, j = 1, 2, 3). Pαβ also depends on the lepton mixing matrix, knownas Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U . U relates the neutrino flavoreigenstates (νe, νµ, ντ ) with neutrino mass eigenstates (ν1, ν2, ν3) in the following wayνeνµ

ντ

=

Ue1 Ue2 Ue3

Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

ν1

ν2

ν3

(2.1)

To explain the neutrino data till date, it is customary to parametrize PMNS matrix U , withthe three mixing angles θ12, θ13, θ23 and one complex phase δ. In terms of these parame-ters the matrix U is represented as (where ’c’, ’s’ stand for cosine and sine of the anglesrespectively),

U =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

(2.2)

In addition one may have two more CP-violating phases, ξ and η, in the neutrino sector ifneutrinos are of Majorana type. These phases show up in very special situations. To relatethe mixing coefficients to experimentally observed effects, it’s very important to indicatethe ’ordering’ of neutrino mass eigenstates. In literature, this is done in the following way: m2

2 > m21 and ∆m2

21 < |∆m232|. So, there are practically three mass-related oscillation

observables : ∆m221 (positive-definite), |∆m2

32| and the sign of ∆m232. A positive (negative)

sign for ∆m232 implies m2

3 > m22 (m2

3 < m22) and refers to what is called normal (inverted)

24

2.2. EXPERIMENTS WITH NEUTRINO

neutrino mass hierarchy. We list below the best fit values of various neutrino oscillationparameters obatined from global analysis of experiments [32]

∆m221 = 7.54× 10−5eV 2, |∆m2

32| = 2.47(2.46)× 10−3eV 2 (2.3)

sin2θ12 = 0.307, sin2θ23 = 0.39, sin2θ13 = 0.0241(0.0244), (2.4)

where the values (values in brackets) correspond to m1 < m2 < m3 (m3 < m1 < m2).Itshould be noted that this ’bi-large’ mixing pattern is very different from that in the quarksector, raising questions about some different physics origin. But, at the same time, viru-ally no information regarding δ nor for the other two CP-violating phases ξ and ζ areavailable. The same can be said about the sign of ∆m2

32. The absolute values of neutrinomasses are not known yet, we only know the differences between mass squared values.From these information, the possibility that the lightest neutrino is virtually massless(< 10−3eV ), can’t be ruled out. There lies another possibility that, all neutrino massesare practically the same (m1 ∼ m2 ∼ m3 ∼ 0.1eV ). Clearly, experiments and searchesoutside the arena of neutrino oscillations are very much required to obtain an estimationof absolute values of neutrino masses.

Several questions regarding neutrino sector still remain unanswered. For example,questions like –what is the ordering of neutrino masses ?, Is there CP violation in the leptonsector ?, Are neutrinos Dirac or Majorana particles?, What is the absolute mass scale for theneutrinos?, Are there indications of new physics in existing experimental data? and lastly, issome physics beyond the SM responsible for the neutrino mass and mixing? are still open [33].Various brilliant experiments have been performed, some are still ongoing and futureexperiments are being designed to obtain more information which will eventually helpto answer these questions. Some of these experiments are listed in the next section.

2.2 Experiments with neutrino

We briefly discuss below some of the experiments which helped to obtain the presentpicture of neutrino masses and mixing :

• Atmospheric Neutrinos : Atmospheric neutrinos are produced when cosmic rays(mostly protons) hit the atmosphere and produce a shower of pions. The pionsthen decay into muons (µ) and muon-neutrinos (νµ). The muons further decay into

25


electrons, electron-neutrinos and muon-antineutrinos. Considering the events fromboth electron-type and muon-type, it is possible to define the ratio ’R’ as [34] :

R ≡ (µ/e)

(µ/e)MC

(2.5)

where ’MC’ denotes the Monte-Carlo predictions. The denominator is the the-oretical estimation for (µ/e) ratio assuming that neutrino interactions are gov-erned by SM and they are massless. If there is no neutrino oscillation, R mustbe 1. Several experiments were performed to study atmospheric neutrino oscilla-tion and thus this ratio. Among these experiments NUSEX [35, 36], Frejus [37–40],Soudan and Soudan-II [41–44], MACRO [45–47] were calorimeter-like detectors,while Kamiokande [48–54], Super-Kamiokande [55–60], IMB [61–63] were waterCherenkov detectors. All these showed a deviation from calculated value for ’R’and thus confirmed neutrino oscillation in atmosphere. These data indicate thatthere is either a depletion of muon-neutrinos or an excess of electron-neutrinos dueto some non-standard property of neutrinos. Any confusion about the evidence ofthis atmospheric neutrino anomaly subsided after the Super-Kamiokande experi-ment measured the atmospheric neutrino flux and they are now considered to bestandard results [64]

∆m2νµντ ' (1.5− 3)× 10−3eV 2 (2.6)

sin22θνµντ > 0.9− 1 (2.7)

The most appealing solution for the atmospheric neutrino anomaly is to assumethat the muon-type neutrinos oscillate into tau-type neutrinos keeping the electron-neutrino flux unaltered. The observed data from Kamiokande as well as Super-Kamiokande experiments support this hypothesis.

• Solar Neutrinos : Thermonuclear reactions that take place inside the core of theSun and produce both photons and neutrinos. A very big step towards experi-mental neutrino physics was obtained when an experiment lead by Ray Davis atthe Homestake mine South Dakota [65], obtained evidence for a solar electron-type neutrino flux. The Homstake experiment was followed by three other ex-periments. Among these three experiments, the Kamiokande (Japan) was a largewater-Cerenkov experiment as already mentioned above and the other two were

26


GALLEX (Italy) [16] and SAGE (USSR/Russia) [18], both of which were radiochem-ical experiments. The last two experiments detected neutrinos via inverse β - decayof Gallium. These three experiments provided complementary information aboutsolar neutrinos. Kamiokande experiment [66–70] was capable of detecting neutri-nos in real time and determine their energy where as the other two experimentshave no such capabilities but they have a lower energy detection threshold.

All these experiments detected solar electron-type neutrino fluxes to be much be-low the values predicted by SM calculations. This was referred to as ’solar neutrinopuzzle’. New light on the solar neutrino puzzle was reflected by the SNO experi-ment [71] results. The hypothesis that supports the data is that although Sun pro-duces only electron-neutrinos, some of these νe’s get converted into other neutrinoflavors i.e neutrino oscillation takes place. Clearly, this is not possible if we stickto the SM framework for neutrinos in which they are massless. One must go be-yond SM to understand this phenomenon. The best fit data for solar neutrinos areas follows [71]

∆m2Solar ' 1.2× 10−5 − 3.1× 10−4eV 2 (2.8)

sin22θSolar ' 0.58− 0.95 (2.9)

• LSND experiment : The LSND (Los Alamos Liquid Scintillation Detector) experi-ment [72] also confirmed neutrino oscillation. In this experiment neutrino oscilla-tions from a stopped muon as well as from a muon in pion decay have been ob-served. The mass squared values and mixing parameters that best fit the data with90% Confidence Level from this experiments are given by [72]

∆m2LSND ' 0.2− 10.0eV 2 (2.10)

sin22θLSND ' 0.003− 0.03 (2.11)

We see that, both the mass-squared differences in equation (2. 3) are much smallerthan ∆m2

LSND. This is sometimes referred to as LSND anomaly. Furthermore, KAR-MEN experiment [73] at the Rutherford laboratory strongly constrained the allowedparameter space of LSND data. The MiniBooNE experiment [74, 75] at Fermilab isalso probing the LSND parameter range to a further extent. None of the proposed

27


solutions to explain the LSND anomaly seem completely satisfactory. It is said that,if LSND observations are confirmed by MiniBooNE experiment, then there is a goodpossibility that a novel physical phenomenon is waiting to be uncovered.

• K2K experiment : After the Super-Kamiokande experiment with atmospheric neu-trinos was performed, K2K experiment [23, 76] was designed to confirm these re-sults. The evidence for neutrino oscillation in Super-Kamiokande was obtained byperforming experiments on neutrinos produced high in the upper atmosphere ofthe earth. In this case, the exact number and energy of the produced neutrinos areuncertain because of the lack of information on the exact energy spectrum and fluxof the cosmic rays hitting earth’s atmosphere. The K2K (KEK to Super-Kamiokande)experiment is the first long-baseline neutrino oscillation experiment which uses ar-tificially produced and controlled neutrino beam. In this experiment, instead ofrelying on neutrinos produced in the upper atmosphere, neutrinos are producedin KEK accelerator facility in Japan. Neutrinos produced in such a way, are thendetected by the Super-Kamiokande detectors.

• KamLAND experiment : The KamLAND (Kamioka Liquid-scintillator Anti-Neutrino Detector) experiment [26, 27, 77, 78] in Japan, stands to be the first exper-iment to find indisputable evidence for neutrino mass using a terrestrial source ofanti-neutrinos. KamLAND detects hundreds of anti-neutrinos per year from nu-clear reactors located few hundred kilometers away, which is considered as a su-perb improvement over previous attempts by any other detector. Since 2002, Kam-LAND has observed anti-neutrino deficit as well as energy spectral distortion, thusconfirming neutrino oscillations and non-zero neutrino masses. The KamLAND ex-periment determined a precise value for the neutrino oscillation parameters ∆m2

21

and θ12 at 3σ level, given by [78] :

∆m221 = 7.59+0.21

−0.21 × 10−5eV 2 (2.12)

tan2θ12 = 0.47+0.06−0.05 (2.13)

This is considered as one of the profound discoveries in neutrino physics.

• Daya Bay experiment : The Daya Bay experiment [30, 31, 79, 80] is another veryimportant neutrino oscillation experiment with the goal to measure the θ13 mixingangle, using anti-neutrinos produced by the reactors at the Daya Bay Nuclear Power

28


Plant via the inverse beta decay process. Correct measurement of θ13 angle is verycrucial, since its magnitude has implications for CP-violation in the leptonic sector.Previously, the most sensitive limit on the value of θ13 was provided by the CHOOZreactor experiment [28] : θ13 < 0.17. The goal of Daya Bay experiment is a measure-ment of sin2θ13 upto 0.01 or better, which is an order of magnitude better sensitivitythan the CHOOZ limit. The experimental set-up at Daya Bay uses larger detectorsto increase statistics and detectors are placed deeper underground to suppress back-ground to achieve improved precision measurements over previous experiments.

• SNO+ experiment : An upadated version of the SNO experiment with solar neu-trinos has been designed and named as SNO+ [81]. The primary goal of SNO+ isa search for the Neutrinoless double β decay process (0νββ). Additionaly SNO+aims to measure reactor anti-neutrino oscillations, low-energy solar neutrinos, geo-neutrinos and to search for exotic physics. The SNO+ is kilo-tonne scale liquid scin-tillator detector, which detects neutrinos when they interact with electrons and nu-clei in the detector and produce charged particles, which in turn produce flashes oflight while passing through the scintillator.

Apart from these experiments mentioned above there are various other experimentslike, MINOS [82,83], IceCube [84], NOνA [85,86], RENO [29], Double CHOOZ [28],JUNO [87], Icarus [88], CNGS [89], LBNE [90], T2K [91], CHIPS [92], RENO-50 [93]and so on, which are trying provide more information and bounds regarding theneutrino masses and mixing.

• Neutrinoless double β decay process (0νββ) : The question whether the neutrinosare Dirac or Majorana particles is one of the important experimental questions. Theanswer to this question will have a profound impact on the theoretical description ofneutrinos. For interesting articles on neutrinoless double β decay process, please seethe incomplete references [94–111]. Also, oscillation experiments described above,only depend on the mass-squared differences of neutrino mass eigenstates and mix-ing angles. Therefore, to have a compact picture of absolute neutrino mass scale, weneed other experiments. Given the current bounds on neutrino masses and mixing,one such experiment, where the lepton number is violated by Majorana neutrino, isneutrinoless double β decay process (0νββ). For a nucleus with atomic number ’Z’,the 0νββ process is represented as : Z −→ (Z + 2) + e− + e−. Clearly this violates

29


lepton number by 2 units. The amplitude for 0νββ process is proportional to [112] :

A0νββ ∝ ΣiU2ei

mi

E≡ mββ

E(2.14)

where Uei are the elements of PMNS mixing matrix, E is the energy of the processandmββ (commonly referred to asmee) is the effective neutrino mass for the process.It’s important to note that A0νββ is directly proportional to the neutrino mass. Thisindicates that, value of mee, not only depends on the mass squared differences andmixing angles but also crucially depends on the magnitude of the masses. That’swhy searches for 0νββ process set bounds on the overall scale of neutrino mass.Also, as mentioned above, 0νββ signifies lepton number violation and thus indi-cates the existence of Majorana-type neutrinos. Till now, there is no firm experi-mental evidence to confirm or overrule this idea. Of course, experimental evidencefor 0νββ process would establish the Majorana nature of neutrinos and thus wouldfavor theoretical models, such as seesaw mechanisms, which tries to explain thesmallness of neutrino mass by assuming neutrinos as Majorana particles. Withinthe neutrino community it is expected that a sharp peak at the ββ end point wouldgive a quantitative estimation of 0νββ decay rate and ultimately provide a measureof effective Majorana mass mββ . Some years ago, a controversial analysis of theHeidelberg-Moscow data [113] announced a positive hint for 0νββ. Their claim wasthat mee lies between 0.2 − 0.6eV , but the questions on absolute scale of neutrinomasses still remain open.

Another important indication on absolute scale of neutrino mass is expected to comefrom the end point searches of tritium decay [114]. This experiment measures theparameter mν ≡

√Σi|Uei|2m2

i . We see that this experiment uses a different combi-nation of mass and mixing parameters than the 0νββ process. The KATRIN experi-ment [115] has been proposed for a high sensitive search for absolute scale of mν . Itis expected that KATRIN will be able to reach a sensitivity of 0.3eV .

2.3 Motivations for physics beyond the SM

By now it is confirmed that neutrinos have non-zero but very tiny masses compared allother fundamental fermion masses in the SM. Two striking features regarding the neu-trino masses really worth noticing that — (i) neutrino masses are at least six orders ofmagnitude smaller than the electron (lightest charged fermion) mass and (ii) there is a’mass-gap’ between the electron mass and the largest allowed value of neutrino mass.

30

2.3. MOTIVATIONS FOR PHYSICS BEYOND THE SM

Neither we know the reasons for neutrino masses being so small, nor we know whythere is such a large gap between mass of neutrino and other charged leptons. In addi-tion to this, there is also the possibility that the massive neutrinos may turn out to be ofMajorana-nature, unlike any other particle in SM framework. As already mentioned, de-termination of the nature of neutrinos i.e Dirac or Majorana-type will have tremendousimpact on the understanding of neutrino physics and particle physics in general. Thismight not only uncover the origin of neutrino masses but will also indicate that conser-vation of lepton number is not really a sacred principle of nature. We have also discussedin the previous section that the most promising way to unravel this puzzle is to look forneutrinoless double beta decay processes.

At this point, various searches and theoretical models are being proposed to find outactually what kind of augmented, new physics scenarios beyond SM leads to non-zeroneutrino masses. There are many possible ways to modify the SM so that small neutrinomass can be accommodated in the framework, with each probable way differing fromone another. Many theoretical models are allowed by the existing data that can hope todescibe small neutrino masses. Therefore, the main point is to find out how the correctmodel for neutrino mass generation can be identified from this jigsaw of models. Per-haps the answer to this question lies with the upcoming neutrino experiments in the nextdecades.

The non-zero neutrino mass has profound theoretical significance. In the SM frame-work, masses of all elementary particles are generated when the neutral component ofHiggs field accqires a vev via electroweak symmetry breaking. Thus the mass scalesof these particles are determined by this vev, v ' 246 GeV. Non-zero neutrino massesmay turn out to be the first signature of a new mass scale, completely unrelated to theto electroweak symmetry breaking scale. Alternatively, this may also indicate that theelectroweak symmetry breaking is actually more complex than dictated by SM.

Apart from this, the pattern of neutrino mixing is also puzzling. A glance at the ab-solute values of the entries of quark mixing matrix i.e CKM matrix shows that the quarkmixing matrix is almost proportional to the identity matrix and the small values of off-diagonal elements there follows a hierarchial pattern. But the entries of lepton mixingmatrix i.e PMNS matrix are given by [33]

|U | '

0.8 0.5 0.2

0.4 0.6 0.7

0.4 0.6 0.7

(2.15)

31


This shows that the PMNS matrix is far from diagonal and follows no hierarchial pattern.Several proposals have been put forward and researches are going on to understand thepossible relationship, if any, between the quark and lepton mixing matrix and to find outthe ordering principle, if any, responsible for the observed pattern of neutrino masses andlepton mixing. As already pointed out in the previous section, several experiments thatinvolve precision neutrino oscillation measurements, are being proposed and carried outto address this flavor puzzle. Improved understanding of neutrino interactions – beyondthe current paradigm is very much needed. Experimental set-ups on neutrino scatteringare expected to be extremely helpful in this regard.

So, from the the experimental observations and facts described in the above para-graphs it must be realized that, we need to seek beyond the paradigm of SM, both theo-retically and experimentally, to be able to find out solutions to the neutrino riddle. Withthis motivation, we present a reference to some of the neutrino mass generation schemesexisting in the literature.

2.4 Models of neutrino mass generation

There exists numerous models that descibe neutrino mass generation consistent with theexisting data. It goes beyond the scope of this dissertation to describe them all. Therefore,we discuss only some of these models briefly here :

• Adding a right-handed neutrino to SM : The simplest possible extension of SMwhich generates non-zero neutrino mass is one where a right-handed neutrino isadded to the SM [116]. In this case both the left-handed (νL) and right-handed (νR)components are present in the scenario and they can form a mass term. This massis similar to the mass terms of other charged leptons and quarks and the mass scaleis therefore governed by the electroweak symmetry breaking scale. The neutrinomass term in this case is given by : mν = Yνv√

2, where Yν is the corresponding yukawa

coupling. So, we see that to obtain sub-eV range neutrino mass, Yν has to be' 10−12

or less. But, the occurance of such a tiny coupling constant in the theory is veryunnatural and a robust theory must give an explanation behind such smallness.

• The seesaw mechanisms : The basic principle behind the seesaw mechanism isthat some extra fermionic or scalar fields are added to the SM. These extra fieldsare assumed to have relatively large mass scale compared to the electroweak scale.In some models the new physics scale is considered to be of the order of 1012 GeV

32

2.4. MODELS OF NEUTRINO MASS GENERATION

or greater. These BSM fields with relatively higher mass scale in turn generatessmall Majorana neutrino masses through the new Yukawa couplings with SM lep-ton fields.

There are three varieties of seesaw mechanism that are commonly used to generatesmall neutrino masses and they are called Type I, Type II and Type III seesaw mech-anism [118–120]. In the case of Type I seesaw, one can add at least two fermionicsinglets Ni (right-handed neutrinos) and the neutrino masses read as: mI

ν 'h2νv

2

MN.

Where hν is Dirac Yukawa coupling, v ' 246 GeV, the SM Higgs vev and MN is theright-handed neutrino mass. Small neutrino masses can be generated by properlyadjusting the parameters hν and MN . Type II seesaw is the case, where the scalarsector of the SM is extended by adding at least one SU(2)L Higgs triplet ∆ with hy-percharge 2. In this scenario neutrino masses are obtained as: mII

ν ' Yνv∆, with v∆

as the vev of the neutral component of the triplet and Yν is yukawa coupling con-stant. The triplet vev and the Yukawa coupling can be adjusted to give the neutrinomass' sub-eV. But we have to keep in mind while choosing the value for triplet vev,that it has to satisfy the ρ - parameter constraint presented in Chapter 1 of this dis-sertation. In Type III seesaw mechanism, by adding at least two extra matter fields(which are normally taken as fermionic triplets) with zero hypercharge, small neu-trino masses can be generated as : mIII

ν ' Γ2νv

2

MΣ, where Γν is Dirac Yukawa coupling,

v, the SM Higgs vev, MΣ is mass of the fermionic triplets. A detailed discussion onseesaw mechanism is presented in the next chapter.

• Inverse seesaw mechanism : Another realization of seesaw mechanism is the so-called ’Inverse Seesaw mechanism (ISS)’ [122]. In this case, small neutrino massesarise as a result of new physics at the TeV scale, which may be probed at the colliderexperiments like the LHC. In the ISS framework, three right-handed neutrinos (NiR)and three extra singlet neutral fermions(SiL) are added to the SM. In addition tothese singlet fields, three left-handed neutrinos (νiL) already exist in the SM. So,there are altogether six singlet fields in the framework now and ISS can be realizedthrough the Lagrangian [123] :

LISS = −Y ijν LiΦNRj − SLjM ij

RNRi −1

2µijS (SLi)

cSLj + h.c (2.16)

This Lagrangian can be re-written as :

LISS = −νiLmijDNRj − SLjM ij

RNRi −1

2µijS (SLi)

cSLj + h.c (2.17)

33


Where L is the left-handed lepton doublet, Φ is SM Higgs doublet, Φ = iσ2Φ∗ withσ2 being the corresponding Pauli matrix, Yν is the 3 × 3 neutrino Yukawa couplingmatrix, MR is a lepton number conserving complex 3× 3 mass matrix, mD is a 3× 3

Dirac mass matrix given by mD = Yνv with Higgs vev v = 246 GeV and µS is acomplex 3 × 3 symmetric Majorana mass matrix, that violates lepton number con-servation by two units. Setting µS to zero, would restore the conservation of leptonnumber and thus would increase the symmetry of the model.

After electroweak symmetry breaking, in the basis (νL, SL, N cL), the 9 × 9 neutrino

matrix is obtained as

Mν =

0 mTD 0

mD 0 MTR

0 MR µS

(2.18)

This mass matrix is symmetric and complex, so it can be diagonalized by a 9 × 9

unitary matrix Uν in the following way

UTνMνUν = Diag(m1,m2, ...m9) (2.19)

This produces three light and six heavy mass eigenstates. In the limit, µS mD MR, the mass matrixMν can be diagonalized in blocks and produces the 3× 3 lightneutrino mass matrix as

Mlight ' mTD(MT

R )−1µSM−1R mD (2.20)

This matrix is then diagonalized using the PMNS matrix to generate masses forthree light neutrinos as

UTPMNSMlightUPMNS = Diag(mν1 ,mν2 ,mν3) (2.21)

Here mν1 ,mν2 ,mν3 are masses of three light neutrinos. The standard neutrinos withmass at sub-eV scale can be obtained for mD at electroweak scale, MR at TeV scaleand µS at the keV scale.

• Radiative corrections : Lastly, in some models involving higher dimensional op-erators, small neutrino mass is generated via radiative corrections of new couplingconstants at loop level. For a deeper understanding on these models, please see theincomplete references [124–129].

34

2.5. TEXTURE ZEROS OF NEUTRINO MASS MATRICES

As mentioned above, apart from these techniques there exist a number of theoreticalmodels in literature that describe tiny neutrino masses consistent with present data. Inthe next section, we turn to describe in brief the textures for neutrino masses that fit ex-perimental data very well and make an attempt to predict the pattern of neutrino masses.

2.5 Texture zeros of neutrino mass matrices

We have seen that the structure of neutrino mass matrices is very important for futurestudy of the underlying symmetries and resulting dynamics. From mixing matrix i.ePMNS matrix U and the mass eigenvalues, it is possible to write down the allowed neu-trino mass matrix for any arbitrary mass pattern assuming that the neutrinos are of Ma-jorana nature. One such way, inspired by the studies of quark mixing matrices, is toconsider mass matrices with zeros in it and at the same time, search for appropriate sym-metries that will establish the viability of zero entries. The usefulness of such an approachis that, because of the presence of fewer parameters in the mass matrices it may be pos-sible to obtain relations between different observables such as masses and mixing angles,that have higher predictiveness and can be tested in experiments. Such family of neutrinomass matrix textures, with independent zero entries in it, are referred to as ’zero textures’.

The simplest symmetries that can be imposed in the context of texture zeros, are con-sidered to be Abelian in nature. Given a mass matrix with texture zeros, it is almostalways possible to find an extended scalar sector and suitable Abelian symmetries, suchthat the texture zeros originate as a result of symmetry requirements on the SM fieldsand extended sector fields. In that sense, texture zeros always can be associated withAbelian symmetries. The most simple scenario in this case takes place when the chargedlepton matrix is diagonal, signifying six texture zeros in it. At the same time, by assumingthat neutrinos are Majorana fermions, some texture zeros can be placed in the symmetricMajorana mass matrices of neutrinos. For an extensive review of ’texture-zeros’, pleasesee [130–132] and references therein.

The most popular among these texture zero approaches is the idea of ’two-zero-textures’i.e the case where the symmetric Majorana neutrino mass matrices have two indepen-dent zero entries. This situation was first studied by Frampton, Glashow and Marfatia in2002 [133]. Out of fifteen possible two-zero-textures, only seven are found to be consistent

35


with the bounds imposed by neutrino experimental data. They are listed below :

Case A1 : Mν ∼

0 0 ×0 × ×× × ×

, Case A2 : Mν ∼

0 × 0

× × ×0 × ×

(2.22)

Case B1 : Mν ∼

× × 0

× 0 ×0 × ×

, Case B2 : Mν ∼

× 0 ×0 × ×× × 0

(2.23)

Case B3 : Mν ∼

× 0 ×0 0 ×× × ×

, Case B4 : Mν ∼

× × 0

× × ×0 × 0

(2.24)

Case C : Mν ∼

× × ×× 0 ×× × 0

(2.25)

It can be easily seen that the seven allowed two-zero textures can be divided into threeclasses, with each class having their own typical implications on neutrino data [133]. Forexample, textures within class A have, Mee = 0, which implies that neutrinoless doublebeta decay (0νββ) is forbidden at least at tree-level for textures of this class. Class C(which is a single - element class) alone allows both the µµ and ττ elements to be zero atthe same time.

Class A textures also imply that the angle θ12 can turn out to be relatively large. On thecontrary, textures of class B allow the 0νββ process. According to class C, the 0νββ decayis likely to be observable and the angle θ12 must be large enough to permit a search forCP-violation. Various efforts have been made to give reasonable structures of neutrinomass matrices by reconciling two-zero textures and experimental neutrino data. Amongthe viable cases classified by Frampton et. al., only A1 and A2 predict θ13 to be differentfrom zero at 3σ. At the best fit value A1 and A2 predict, 0.024 ≥ sin2θ13 ≥ 0.012 and0.014 ≤ sin2θ13 ≤ 0.032 respectively. The cases B1, B2, B3 and B4 predict nearly maxi-mal CP-violation i.e cosδ ≈ 0. This is an important result predicted by two-zero textureof Majorana neutrino mass matrix in the light of MINOS, T2K and K2K experimentalresults, which gave hints for a large θ13 value [134]. Fritzsch et. al. [135] discusses an in-teresting observation that given the values of three flavor mixing angles (θ12, θ23, θ13) andtwo neutrino mass-squared differences (∆m2

12,∆m223), it is in principle possible to fully

determine three CP-violating phases (δ, ζ, ξ) and three neutrino masses. Recently, Meloni

36

2.5. TEXTURE ZEROS OF NEUTRINO MASS MATRICES

et. al. [136] analyzed some of the two zero textures of neutrino masses in the light of thePlanck data, that put a quite stringent upper bound on the sum of the active neutrinomasses : Σ ≡ Σimνi < 0.23 eV with 95% CL. They have shown that, texture C is notcompatible with the normal ordering of the neutrino mass eigenstates. The process ofanalyzing predictions from two-zero textures of neutrino mass matrices and examiningthem in the light of ever increasing experimental data is still going on. For some moreinteresting articles on ’two-zero textures’ and ’zero-textures’ in general, please see theincomplete reference [137–143]. It can be said without hesitation that two-zero texturesand texture zero approach in general, serve as a guideline to provide a viable model ofneutrino mass structure.

37


38

Chapter 3

Seesaw mechanism

3.1 Introduction to seesaw

We have already mentioned seesaw mechanism briefly in the previous chapter. Here wepresent a more elaborate discussion of the seesaw feameworks, which generate Majoranamasses for neutrinos. Such a mass term is generally of the form

Lm =1

2mLνL(νL)c (3.1)

where νL is a left-handed neutrino field and (νL)c = CνLT , C = iγ2γ0 being the charge

conjugation operator. Such Majorana masses are possible since both the neutrino andthe anti-neutrino have zero electric charge. Majorana masses for neutrinos are then notforbidden by electric charge conservation. However, such masses for neutrinos violateslepton number conservation by two units. Also, νL possesses non-zero isospin and hy-percharge. So in the SM framework, assuming only Higgs doublets, such Majorana massterms for left-handed neutrinos are forbidden by gauge invariance. The idea of the seesawmechanism is to generate such terms effectively, after heavy external fields are integratedout. Ultra-small neutrino maases then can be justified in terms of ’big’ mass parametersof these heavy fields.

There are three different realizations of seesaw mechanism. They are all based uponthe fact that, according to group theory, two doublets can be decomposed into a tripletand a singlet (2 ⊗ 2 ≡ 3 ⊕ 1) combination. Thus the left-handed lepton doublet and theHiggs doublet of SM can couple either to a triplet or a singlet field. The three seesawmechanisms outlined here bring such ’big’ mass parameters from three different sectors,each of them going beyond the SM in its own way.

39

CHAPTER 3. SEESAW MECHANISM

3.2 Type I seesaw : Fermion singlets

To understand the principle of type I seesaw mechanism [118], let’s start with the neutrinomass matrix once more. We assume that right-handed neutrinos (νR) are present in thescenario in addition to the usual left-handed neutrinos (νL) of the SM. Therefore, it is nowpossible to construct a Dirac mass term for the neutrinos as follows

LDiracMass = mDνRνL + h.c (3.2a)

=1

2(mDνRνL +mD(νL)c(νR)c) + h.c . (3.2b)

In general Majorana type mass terms are also possible for neutrinos, since they have zeroelectric charge. The Majorana mass term for left and right-handed neutrinos are writtenas

LLMass =1

2mL(νL)cνL + h.c (3.3a)

LRMass =1

2mR(νR)cνR + h.c (3.4a)

where ’L’ and ’R’ stands for left and right-handedness respectively. Let us define a vectorfield nL such that

nL =

(νL

(νR)c

), (nL)c = ((νL)cνR) (3.5)

where (nL)c is the charge conjugate field of nL. Then the total mass lagrangian for neutri-nos can be written as

LtotalMass = LDiracMass + LLMass + LRMass (3.6)

=1

2(nL)cMnL (3.7)

The mass matrix M , whose diagonalisation gives the physical neutrino masses is givenby

M =

(mL mD

mTD mR

)(3.5)

The positive mass eigenstates of this matrix are

m1,2 =1

2(mL +mR ±

√(mL −mR)2 + 4m2

D) (3.6)

40

3.2. TYPE I SEESAW : FERMION SINGLETS

In the seesaw scenario the added right-handed neutrinos are assumed to have highermass well above the electroweak symmetry breaking (EWSB) scale, while the Dirac massmD (like other charged fermions) is considered to be of the order of EWSB scale. There-fore, in this scenario, mR mD. On diagonalisation of M , the eigenvalues correspondingto the neutrino mass eigenstates are obtained as

m1 'm2D

mR

(3.7)

m2 ' mR (3.8)

It is clear from these equations that, now we have a right-handed neutrino with massscale ΛN = mR, which also may be the scale of some new physics and at the same time avery light neutrino, with mass suppressed by mD

ΛN. This new mass scale is often taken to

be close to the grand unification scale to explain the proposed sub-eV masses of the threeleft-handed neutrinos. In fact, heavier the new right-handed neutrino, the lighter is theleft-handed neutrino. This is the underlying basic principle of seesaw mechanism.

Formally, in type I seesaw, right-handed heavy SU(2) singlet fermion fields (NR) withzero hypercharge are added to the SM fields to produce non-zero neutrino masses. Theleft-handed lepton doublets then couple to the Higgs field and the newly introducedright-handed fermion fields to produce the Majorana mass terms for the neutrinos. Theextra piece of lagrangian for this new heavy right-handed fields is given by [121]

LNew = iNR/∂NR − [1

2NRMNN

cR + lLΦY †NNR + h.c] (3.9)

where the first two terms represent the kinetic term and Majorana mass term for the right-handed fermion fields and the third one is the Yukawa interaction term. lL and Φ are theleft-handed lepton doublet and Higgs field respectively. The vev of Higgs field is denotedby ’v’.

The left-handed neutrino mass term can be estimated from the above lagrangian. Thelagrangian can be solved to get the equation of motion. Expansion of the propagatorgives a factor i

/p−MN, which for low momenta p can be approximated by (−i)

MN, with MN

as the mass of heavy right-handed fermion fields. Then integrating out the heavy fieldsusing the equation of motion, namely,

∂LNew

∂NR

= 0 (3.10)

41


one obtains the effective dimension 5 operator [121]

δLd=5 =1

2cd=5αβ (lcLαΦ∗)(Φ†lLβ) + h.c (3.11)

The coefficient cd=5 is given by [121]

cd=5 = Y TN

1

MN

YN (3.12)

After electroweak symmetry breaking, the neutrino mass is obtained by inserting theHiggs vev ’v’ and is expressed as [121]

mν =v2

2cd=5 = Y T

N

v2

2MN

YN (3.13)

From this relation, light Majorana neutrino mass can be generated by properly adjustingthe value of the parameters YN and MN .

3.3 Type II seesaw : Scalar triplets

In this case [119] the scalar sector of the SM is extended by adding at least one SU(2)

scalar triplet ∆(∆1,∆2,∆3) with hypercharge 2. The electromagnetic charged states forthe triplet can be obtained from the couplings to the leptons

¯lL(τ ·∆)lL = (−(eL)c(νL)c)

(∆3 ∆1 − i∆2

∆1 + i∆2 −∆3

)(νL

eL

)= −(eL)c∆3νL − (eL)c(∆1 − i∆2)eL + (νL)c(∆1 + i∆2)νL − (νL)c∆3eL

(3.14)

Since eL and (eL)c have charge -1 and νL, (νL)c have no charge, the charged states of thetriplet are given by [121]

∆++ ≡ 1√2

(∆1 − i∆2), ∆+ ≡ ∆3, ∆0 ≡ 1√2

(∆1 + i∆2) (3.15)

The Yukawa interaction term of the new triplet, which violates lepton number by twounits, is represented as

Lyuk = Y∆¯lL(τ ·∆)lL (3.16)

where ¯lL = (−(eL)c(νL)c), τ ’s are the Pauli matrices and Y∆ is the Yukawa coupling matrix.The rest of the part of the lagrangian of the triplet field containing the scalar potential

42

3.3. TYPE II SEESAW : SCALAR TRIPLETS

terms is given by [121] ,

L∆ = M2∆∆†∆ +

1

2λ2(∆†∆)2 + λ3(Φ†Φ)(∆†∆)+

1

2λ4(∆†T i∆)2 + λ5(∆†T i∆)(Φ†τ iΦ)+

(µ∆Φ†(τ ·∆)†Φ + h.c)

(3.17)

Here, T i’s are the generators of the triplet representation of SU(2)

T1 =

0 0 0

0 0 −i0 i 0

. T2 =

0 0 i

0 0 0

−i 0 0

, T3 =

0 −i 0

i 0 0

0 0 0

(3.18)

In principle, one can start with gauge invariant as well as lepton number conservingtriplet Yukawa interactions by assigning lepton number L = −2 to the scalar triplet. Lcould then be broken spontaneously once ∆0 acquires a vev [144]. However, that wouldhave led to a massless SU(2) triplet Goldstone boson with unsuppressed coupling withthe Z boson. The consequence is a much larger contribution to the invisible decay widthof the Z than is permitted by the LEP data. Therefore, explicit L-violation is phenomeno-logically preferable in this scheme. A way to generate the triplet vev is through the tri-linear interaction term in the scalar potential. In this case, minimization of the scalarpotential produces a vev (v∆) for the neutral component of the triplet ∆0, when the Higgsdoublet Φ acquires a vev and is given by [116]

v∆ ≡µ∆v

2

M2∆

(3.19)

It can be seen that, triplet vev can be made small by choosing a large value for the massof the triplet field or by choosing the coefficient of the trilinear term to be vary small.Keeping a small value for the triplet vev (v∆) is crucial in this model since it is directlyrelated to neutrino masses through the Yukawa coupling term. Another important reasonto keep the triplet vev small is to respect the ρ-parameter constraint, which restricts [145]

ρ = 1.0008+0.0017−0.0007 (3.20)

Since the neutral component of triplet ∆0 couples to the W and Z-bosons, its vev con-tributes to their masses. Instead of the SM relations, we now have [34]

M2W =

1

4g2

2(v2 + 2v2∆) (3.21)

43


M2Z =

1

4(g2

1 + g22)(v2 + 4v2

∆) (3.22)

so that, expression for ρ-parameter now reads as [34],

ρ ≡ M2W

M2Zcos

2θW=

1 + 2v2∆/v

2

1 + 4v2∆/v

2(3.23)

Then using the constraint on ρ-parameter, we obtain an upper bound for the allowedvalue of v∆ as [34],

v∆

v< 0.07 (3.24)

Majorana masses of neutrinos are generated when the neutral component of the triplet∆0 acquires vev and the relation between the neutrino mass, Yukawa coupling and tripletvev is expressed as :

M ijν = v∆Y

ij∆ (3.25)

Here the coupling Y ij∆ decide the pattern of neutrino mixing, in analogy to the right-

handed neutrino mass matrix in type I seesaw. The distinctive feature of type II seesawlies in a different vev being responsible for neutrino masses, as compared to the otherfermion masses. Thus the essence of this scenario is v∆ v, thanks to the character of thescalar potential.

3.4 Type III seesaw : Fermion triplets

Type III sesaw [120] is very much similar to the type I seesaw mechanism. In the type IIIseesaw framework at least two heavy SU(2) right-handed fermion triplets are added in-stead of fermion singlet fields. This is because at least two non-vanishing neutrino masseshave to be generated in order to explain the oscillation data. The fermion triplet fields Σ

(Σ1,Σ2,Σ3) have zero hypercharge. The Majorana mass term as well as the dynamics ofthe fields is governed by the lagrangian [121]

LΣ = iΣR /DΣR − [1

2ΣRMΣΣc

R + ΣRYΣ(Φ†τ lL) + h.c] (3.26)

The covariant derivative of the above equation is given by

Dµ = ∂µ − ig1Y

2Bµ − ig2

T a

2W aµ , (3.27)

44

3.4. TYPE III SEESAW : FERMION TRIPLETS

Where, T a’s are the 3×3 generators of SU(2) introduced in the previous section. YΣ is theYukawa coupling and lL,Φ are the SM lepton and Higgs doublets. The relation betweenthe SU(2) components of Σ and the electric charge eigenstates are as follows [121]

Σ± ≡ Σ1 ∓ iΣ2√2

, Σ0 ≡ Σ3 (3.28)

In the case of type III seesaw also, the Lagrangian can be solved to get equation of motionfor the fields. Following the same procedure as in the case of type I seesaw mechanism,after expanding the propagator and integrating out the heavy fields, the effective fivedimensional operator is obtained as [121]

δLd=5 =1

2cd=5αβ ( ¯lLατΦ)(Φ†τ lLβ) + h.c (3.29)

where the coefficient is [121]

cd=5 = Y TΣ

1

MΣ

YΣ (3.30)

Majorana neutrino masses are obtained when the Higgs field acquires a vev and is givenby [121]

mν =v2

2Y T

Σ

1

MΣ

YΣ (3.31)

It has been already pointed out that type III seesaw is very much similar to type I seesawmechanism. We comment on some general features of these two models for neutrinomass generation. In both the cases, the effect of heavy new degrees of freedom (singletand triplet fields in the case of type I and type III respectively) in low energy phenomenacan be manifested by adding higher-dimensional operators to the SM. It is well knownthat, given the SM gauge symmetries and particle content, only one type of dimension-five operator is allowed and its generic form is given by [117]

1

Λ(LΦ)(LΦ) + h.c ≡ v2

Λνν + h.c (3.32)

All other operators that can be constructed are of dimension six or higher. In the aboverelation L and Φ are SM lepton and Higgs doublets respectively, ’v’ is vev of the neutralcomponent of Higgs and reflects the fact that neutrino mass is generated through thisoperator after electroweak symmetry is broken. The most interesting general feature ofthe seesaw models is the existence of a new physics scale Λ, which can be identifiedwith the mass scale of heavy fermion triplet(singlet) MΣ (MN ) of type III (type I) seesawmechanism. For Λ ∼ 1015 GeV or GUT scale, it is possible to obtain sub-eV range masses

45


for light neutrinos in agreement with the current experimental data. Such a high value ofthis new mass scale certainly motivates a possibility that the higher dimensional operatoris indeed generated by some new physics beyond SM.

Of all the three seesaw scenarios described above, the works presented in this thesisare based on type II and type III seesaw framework.

3.5 Two-zero texture and the inadequacy of a single triplet

In the previous chapter we have seen that strong evidence has been accumulated infavour of neutrino oscillation from the solar, atmospheric, reactor and accelerator neu-trino experiments over the last few years. A lot, however, is yet to be known, includingthe mass generation mechanism and the absolute values of the masses, in addition tomass-squared differences which affect oscillation rates. Also, a lot of effort is on to ascer-tain the nature of neutrino mass hierarchy, including the signs of the mass-squared differ-ences. We have also pointed out that a gateway to information of the above kinds is thelight neutrino mass matrix, in a basis where the charged lepton mass matrix is diagonal.In this context we have discussed a possibility that frequently enters into such investiga-tions is one where the mass matrix has some zero entries, perhaps as the consequence ofsome built-in symmetry of lepton flavours. At the same time such ‘zero textures’ lead to ahigher degree of predictiveness and inter-relation between mass eigenvalues and mixingangles by virtue of having fewer free paramaters. We have also pointed out that two-zerotextures have a rather wide acceptability. It has been hinted in [146] that none of the sevenpossible two-zero-texture cases can be achieved by assuming only one scalar triplet. Wewrite the allowed seven two-zero textures for the 3× 3 symmetric Majorana mass matrixof the light neutrinos, denoted by Mν once more for the sake of completeness :

Case A1 : Mν ∼

0 0 ×0 × ×× × ×

, Case A2 : Mν ∼

0 × 0

× × ×0 × ×

(3.33)

Case B1 : Mν ∼

× × 0

× 0 ×0 × ×

, Case B2 : Mν ∼

× 0 ×0 × ×× × 0

(3.34)

Case B3 : Mν ∼

× 0 ×0 0 ×× × ×

, Case B4 : Mν ∼

× × 0

× × ×0 × 0

(3.35)

46

3.5. TWO-ZERO TEXTURE AND THE INADEQUACY OF A SINGLE TRIPLET

Case C : Mν ∼

× × ×× 0 ×× × 0

(3.36)

These textures are defined in a basis where the charged lepton mass matrix is diagonal. Inthe context of Type-II seesaw the Yukawa couplings of scalar triplets ∆k are given by [147]:

LY =1

2

2∑k=1

y(k)ij L

Ti C−1iτ2∆kLj + h.c., (3.37)

where i, j = e, µ, τ , C is the charge conjugation matrix, the y(k)ij are the symmetric Yukawa

coupling matrices of the triplets ∆k. From the above Lagrangian, the neutrino mass ma-trix is obtained as [147]

Mνij = wky

kij (3.38)

withwk s being the vev of the triplets. Among the neutrino mass terms, some are allowed,while others are not, as the consequence of a particular texture. This fact can be associatedwith a conserved global U(1) symmetry, under which all fields have some charge. Underthis symmetry, the lepton doublets Li and the scalar triplet ∆k transform as [147]

Li −→ piLi and ∆k −→ p0∆k (3.39)

with phase factors |pi|, |p0| = 1. An examination of each of the allowed textures revealsthat the three phase factors for different lepton flavors i.e, pe, pµ, pτ have to be differentfrom each other. The Higgs doublet transforms trivially under the horizontal symmetry,thus enabling the charged-lepton mass matrix to be diagonal. Now, let us look at theconsequence of such a symmetry when just one triplet is present. We shall see that thisassuption leads us to a contradiction.

In all the seven possible two-texture-zero cases, the µτ element of Mν is non-zero.Thus, the corresponding Yukawa coupling element yµτ must be non-zero, and the result-ing interaction term must conserve the U(1) charge. This implies [147]

p0pµpτ = 1 (3.40)

We first examine the Cases B1, B2, B3, B4 and C. For these five cases, yee 6= 0. Therefore,upon applying the symmetry operation we have,

p0p2e = 1 (3.41)

47


The inequality of the U(1) charges for the different neutrino flavour eigenstates then re-sults in yeµ picking up a phase factor [147] :

p0pepµ =pµpe6= 1 (3.42)

This leads to the conclusionyeµ = 0. (3.43)

Proceeding in the same way with the eτ element of Yukawa coupling, we obtain

p0pepτ =pτpe6= 1 (3.44)

which again impliesyeτ = 0 (3.45)

On the other hand, it is seen that in none of the cases B1, B2, B3, B4 and C , the Yukawacouplings yeµ and yeτ are both zero. Thus none of these five textures is viable when onlyone triplet is present in the scenario.

We next address the two remaining cases, namely A1 and A2 . For both of these, onehas yµµ 6= 0 . Thus we have agin after the symmetry operation,

p0p2µ = 1 (3.46)

However, that would again mean [147]

p0pµpτ =pτpµ6= 1 (3.47)

This in turn destroys the viablity of these two texture as well. Thus one is forced toconclude that none of the seven possible two-zero-texture cases can be achieved by anabelian horizontal symmetry assuming only one scalar triplet. But, when two or moretriplets are present, then there will be more freedom in terms of the charges possessedby them, and the phase factor relations will be less constraining. Thus the contradictionsthat appear with a single triplet can be avoided, so that at least some of the seven possibletwo-zero textures are allowed. Therefore it is important to examine the phenomenologicalconsequences of an augmented triplet sector if Type-II seesaw has to be consistent withtwo-zero textures. We proceed in that direction in the following chapters.

48

Chapter 4

Doubly charged scalar decays in a type IIseesaw scenario with two Higgs triplets

4.1 Introduction

It is by and large agreed that the Large Hadron Collider (LHC) experiments have dis-covered the Higgs boson predicted in the standard electroweak theory, or at any rate aparticle with close resemblance to it [10, 11]. At the same time, driven by both curios-ity and various physics motivations, physicists have been exploring the possibility thatthe scalar sector of elementary particles contains more members than just a single SU(2)

doublet. A rather well-motivated scenario often discussed in this context is one contain-ing at least one complex scalar SU(2) triplet of the type (∆++,∆+,∆0). We have alreadyseen how a small vacuum expectation value of the neutral member of the triplet, con-strained as it is by the ρ-parameter, can lead to Majorana masses for neutrinos, drivenby ∆L = 2 Yukawa interactions of the triplet. Such mass generation does not requireany right-handed neutrino, and this is the quintessential principle of the type II seesawmechanism [119, 148, 149].

A phenomenologically novel feature of this mechanism is the occurrence of a doubly-charged scalar. Its signature at TeV scale colliders is expected to be seen, if the tripletmasses are not too far above the electroweak symmetry breaking scale. The most conspic-uous signal consists in the decay into a pair of same-sign leptons, i.e. ∆++ → `+`+. Thesame-sign dilepton invariant mass peaks resulting from this make the doubly-chargedscalar show up rather conspicuously. Alternatively, the decay into a pair of same-sign Wbosons, i.e. ∆++ → W+W+, is dominant in a complementary region of the parameterspace, which — though more challenging from the viewpoint of background elimination— can unravel a doubly-charged scalar [150].

49

CHAPTER 4. DOUBLY CHARGED SCALAR DECAYS IN A TYPE II SEESAW SCENARIOWITH TWO HIGGS TRIPLETS

In this chapter, we shall discuss the situation where a third decay channel, namely adoubly-charged scalar decaying into a singly-charged scalar and a W of the same sign, isdominant or substantial. Such a decay mode is usually suppressed, since the underlyingSU(2) invariance implies relatively small mass splitting among the members of a triplet.However, when several triplets of similar nature are present and mixing among them isallowed, a transition of the above kind is possible between two scalar mass eigenstates.Apart from being interesting in itself, several scalar triplets naturally occur in models forneutrino masses and lepton mixing based on the type II seesaw mechanism. In particular,it has been hinted that in such a scenario realization of viable neutrino mass matrices withtwo texture zeros [133,151],using symmetry arguments [152], requires two or three scalartriplets [146]. Here, we take up the case of two coexisting triplets. We demonstrate thatin such cases one doubly-charged state can often decay into a singly-charged state and aW of identical charge. This is not surprising, because each of the two erstwhile studieddecay modes is controlled by parameters that are rather suppressed. In the case of ∆++ →`+`+, the amplitude is proportional to the ∆L = 2 Yukawa coupling, while for ∆++ →W+W+, it is driven by the triplet vacuum expectation value (vev). The restrictions fromneutrino masses as well as precision electroweak constraints make both of these ratesrather small. On the other hand, in the scenario with two scalar triplets with chargedmass eigenstates H++

k and H+l (k, l = 1, 2), the decay amplitude for H++

1 → H+2 W

+,if kinematically allowed, is controlled by the SU(2) gauge coupling. Therefore, if oneidentifies regions of the parameter space where it dominates, one needs to devise newsearch strategies at the LHC [154], including ways of eliminating backgrounds.

We note that the mass parameters of the two triplets, on which no phenomenologicalrestrictions exist, are a priori unrelated and, therefore, as a result of mixing between thetwo triplets, the heavier doubly-charged state can decay into a lighter, singly-chargedstate and a real W over a wide range of the parameter space. In that range it is expectedthat this decay channel dominates for the heavier doubly-charged state. By choosing anumber of benchmark points, we demonstrate that this is indeed the case. In the nextsubsection we begin by presenting a summary of the model with a single triplet andexplain why the decay ∆++ → ∆+W+ is disfavoured there.

4.2 The scenario with a single triplet

In this section we perform a quick recapitulation of the scenario with a single triplet field,in addition to the usual Higgs doublet φ, using the notation of [155]. The Higgs triplet

50

4.2. THE SCENARIO WITH A SINGLE TRIPLET

∆ = (∆++,∆+,∆0) is represented by the 2× 2 matrix

∆ =

(∆+

√2∆++

√2∆0 −∆+

). (4.1)

The vevs of the doublet and the triplet are given by

〈φ〉0 =1√2

(0

v

)and 〈∆〉0 =

(0 0

w 0

), (4.2)

respectively. Thus, the triplet vev is obtained as 〈∆0〉 = w/√

2 . The only doublet-dominated physical state that survives after the generation of gauge boson masses is aneutral scalar H .

The most general scalar potential involving φ and ∆ can be written as [155]

V (φ,∆) = a φ†φ+b

2Tr (∆†∆) + c (φ†φ)2 +

d

4

(Tr (∆†∆)

)2

+e− h

2φ†φTr (∆†∆) +

f

4Tr (∆†∆†) Tr (∆∆)

+ hφ†∆†∆φ+(t φ†∆φ+ h.c.

), (4.3)

where φ ≡ iτ2φ∗. For simplicity, we assume both v and w to be real and positive, which

requires t to be real as well. In other words, all CP-violating effects are neglected in thisanalysis.

The choice a < 0, b > 0 ensures that the primary source of spontaneous symmetrybreaking lies in the vev of the scalar doublet. We assume the following orders of magni-tude for the parameters in the potential:

a, b ∼ v2; c, d, e, f, h ∼ 1; |t| v. (4.4)

Such a choice is motivated by

(a) proper fulfillment of the electroweak symmetry breaking conditions,

(b) the need to have w v small due to the ρ-parameter constraint,

(c) the need to keep doublet-triplet mixing low in general, and

(d) the urge to ensure perturbativity of all quartic couplings.

51


The mass Lagrangian for the singly-charged scalars in this model is given by

L±S = −(H−, φ−)M2+

(H+

φ+

)(4.5)

with

M2+ =

((q + h/2)v2

√2v(t− wh/2)

√2v(t− wh/2) 2(q + h/2)w2

)and q =

|t|w. (4.6)

It is interesting to note that the determinant of this mass matrix will vanish only if t isnegative. The field φ+ is the charged component of the doublet scalar field of the StandardModel (SM). One of the eigenvalues of this matrix is zero corresponding to the Goldstoneboson which gives mass to theW boson. The mass-squared of the singly-charged physicalscalar is obtained as

m2H+ =

(q +

h

2

)(v2 + 2w2), (4.7)

whereas the corresponding expression for the doubly-charged scalar is

m2∆++ = (h+ q)v2 + 2fw2. (4.8)

Thus, in the limit w v, we obtain

m2∆++ −m2

H+ 'h

2v2. (4.9)

It is obvious from equation 4.9 that a substantial mass splitting between ∆++ and H+

is in general difficult. This is clear from Figure 4.1 where we plot the mass differencebetween the two states for different values of h. Sufficient splitting, so as to enable thedecay ∆++ → H+W+ to take place with appreciable branching ratio, will require h ' 1,m∆++ . 250 GeV and a correspondingly smaller mH+ . The limits from LEP and Tevatrondisfavour triplet states with such low masses. Thus one concludes that the phenomenonof the doubly-charged scalar decaying into a singly-charged one and aW is very unlikely.

52

4.3. A TWO HIGGS TRIPLET SCENARIO

0

10

20

30

40

50

60

70

80

90

100

110

200 300 400 500 600 700 800

Mass d

iffe

rence (

GeV

)

Doubly-charged scalar mass (GeV)

Variation of mass difference between doubly and singly charged scalars

h = 1.0h = 0.5h = 0.1

Figure 4.1: Variation of mass difference between the doubly and singly-charged scalars,for various values of the parameter h.

4.3 A two Higgs triplet scenario

There may, however, be some situations where a single triplet is phenomenologicallyinadequate. This happens, for example, when one tries to impose texture zeros in theneutrino mass matrix within a type II seesaw framework by using Abelian symmetries. Ithas been discussed in the previous chapter how a two-zero texture becomes inconsistentwith this framework if a single scalar triplet is present. Having this is in mind, we ventureinto a model consisting of one complex doublet and two Y = 2 triplet scalars ∆1, ∆2, bothwritten as 2×2 matrices:

∆1 =

(δ+

1

√2δ++

1√2δ0

1 −δ+1

)and ∆2 =

(δ+

2

√2δ++

2√2δ0

2 −δ+2

). (4.10)

The vevs of the scalar triplets are given by

〈∆1〉0 =

(0 0

w1 0

)and 〈∆2〉0 =

(0 0

w2 0

). (4.11)

The vev of the Higgs doublet is given by equation (4.2).

53


The scalar potential in this model involving φ, ∆1 and ∆2 can be written as [156]

V (φ,∆1,∆2) =

a φ†φ+1

2bkl Tr (∆†k∆l) + c(φ†φ)2 +

1

4dkl

(Tr (∆†k∆l)

)2

+1

2(ekl − hkl)φ†φTr (∆†k∆l) +

1

4fkl Tr (∆†k∆

†l ) Tr (∆k∆l)

+hkl φ†∆†k∆lφ+ g Tr (∆†1∆2) Tr (∆†2∆1) + g′ Tr (∆†1∆1) Tr (∆†2∆2)

+(tk φ

†∆kφ+ h.c.), (4.12)

where summation over k, l = 1, 2 is understood. This potential is not the most generalone, as we have omitted some of the quartic terms. This is justified in view of the scopeof this thesis, as laid out in the introduction. Moreover, due to the smallness of the tripletvevs, the quartic terms are not important numerically for the mass matrices of the scalars.

As in the case with a single triplet, we illustrate our main point here taking all the vevsv, w1, w2 as real and positive, and with real values for t1, t2 as well. Again, the followingorders of magnitude for the parameters in the potential are assumed:

a, bkl ∼ v2; c, dkl, ekl, hkl, fkl, g, g′ ∼ 1; |tk| v. (4.13)

We also confine ourselves to cases where w1, w2 v, keeping in mind the constraint onthe ρ-parameter.

In general, the scalar potential (4.12) can only be treated numerically. However, sincethe triplet vevs wk are small (we will have wk . 1 GeV in our numerical part), it shouldbe a good approximation to drop the quartic terms in the scalar triplets. In the follow-ing we will discuss the vevs and the mass matrices of the doubly and singly-chargedscalars in this approximation, so that our broad conclusions are transparent. However,the numerical results presented in section 4.4 are obtained using the full potential (4.12),including even the effects of the small triplet vevs. We find that the results are in verygood accordance with the approximation.

For the sake of a convenient notation we define the following 2 × 2 matrices and vec-tors [156]:

B = (bkl), E = (ekl), H = (hkl), t =

(t1

t2

), w =

(w1

w2

). (4.14)

With this notation the conditions for a stationary point of the potential are given by [156](B +

v2

2(E −H)

)w + v2 t = 0, (4.15)

a+ cv2 +1

2wT (E −H)w + 2 t · w = 0, (4.16)

54

4.3. A TWO HIGGS TRIPLET SCENARIO

where we have used the notation t · w =∑

k tkwk. These two equations are exact if oneneglects all terms quartic in the triplet vevs in V0 ≡ V (〈φ〉0, 〈∆〉0). In equation (4.16) wehave already divided by v, assuming v 6= 0. Using equation (4.15), the small vevs wk areobtained as [156]

w = −v2

(B +

1

2v2(E −H)

)−1

t. (4.17)

Now we discuss the mass matrices of the charged scalars. A glance at the scalar poten-tial equation (4.12)—neglecting quartic terms in the triplet scalars—reveals that the firsttwo lines of V make no difference between the singly and doubly-charged scalars. Thus,the difference in the respective mass matrices originates in the terms of the third line. Themass matrix of the doubly-charged scalars is obtained as [156]

M2++ = B +

v2

2(E +H) . (4.18)

As for the singly-charged fields ∆+k , one has to take into account of the fact that they can

mix with φ+ of the Higgs doublet. Writing the mass term as [156]

−L±S =(δ−1 , δ

−2 , φ

−)M2+

δ+1

δ+2

φ+

+ h.c., (4.19)

equation (4.12) leads to [156]

M2+ =

(B + v2

2E

√2v (t−Hw/2)√

2v (t−Hw/2)† a+ cv2 + 12wT (E +H)w

). (4.20)

Obviously, this mass matrix has to have an eigenvector with eigenvalue zero which cor-responds to the would-be-Goldstone boson. Indeed, using equations (4.15) and (4.16), wefind [156]

M2+

(vT

v/√

2

)= 0, (4.21)

which serves as a consistency check.Note that the matrix B largely controls the mass of the triplet scalars and the order

of magnitude of its elements (or of its eigenvalues) is expected to be a little above theelectroweak scale, represented by v ' 246 GeV. On the other hand, the quantities tk triggerthe small triplet vevs, so they should be considerably smaller than the electroweak scale.Therefore, in a rough approximation one could neglect the tk and the triplet vevs in the

55


mass matrix M2+. In that limit, also a + cv2 = 0 and the charged would-be-Goldstone

boson consists entirely of φ+, without mixing with the δ+k .

The mass matrices (4.18) and (4.20) are diagonalized by [156]

U †M2++U = diag (M2

1 ,M22 ) and V †M2

+V = diag (µ21, µ

22, 0), (4.22)

respectively, with [156](δ++

1

δ++2

)= U

(H++

1

H++2

),

δ+1

δ+2

φ+

= V

H+1

H+2

G+

. (4.23)

We have denoted the fields with definite mass by H++k and H+

k , and G+ is the chargedwould-be-Goldstone boson.

The gauge Lagrangian relevant for the decays considered is given by [156]

Lgauge = ig2∑

k=1

[δ−k (∂µδ++

k )− (∂µδ−k ) δ++k

]W−µ

− g2

√2

2∑k=1

wkW−µ W

−µδ++k + h.c. (4.24)

Here g is the SU(2) gauge coupling constant. Inserting equation (4.23) into this La-grangian allows us to compute the decay rates of H++

1 → H+2 W

+ and H++k → W+W+

(k = 1, 2).The ∆L = 2 Yukawa interactions between the triplets and the leptons are [156]

LY =1

2

2∑k=1

h(k)ij L

Ti C−1iτ2∆kLj + h.c., (4.25)

where C is the charge conjugation matrix, the h(k)ij are the symmetric Yukawa coupling

matrices of the triplets ∆k, and the i, j are the summation indices over the three neutrinoflavours.1 The Li denote the left-handed lepton doublets.

The neutrino mass matrix is generated from equation (4.25) when the triplets acquirevevs [156]:

(Mν)ij = h(1)ij w1 + h

(2)ij w2. (4.26)

This connects the Yukawa coupling constants h(1)ij , h(2)

ij and the triplet vevs w1, w2, oncethe neutrino mass matrix is written down for a particular scenario. In our subsequentcalculations, we proceed as follows:

1We assume the charged-lepton mass matrix to be already diagonal.

56

4.4. BENCHMARK POINTS AND DOUBLY-CHARGED SCALAR DECAYS

First of all, the neutrino mass eigenvalues are fixed according to a particular type ofmass spectrum. We illustrate our points by resorting to normal hierarchy of the neutrinomass spectrum and setting the lowest neutrino mass eigenvalue to zero. Furthermore,using the observed central values of the various lepton mixing angles, the elements of theneutrino mass matrix Mν can be found by using the equation

Mν = UMνU†, (4.27)

where U is the PMNS matrix given by [145]

U =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13e

iδ c23c13

(4.28)

and Mν is the diagonal matrix of the neutrino masses. In equation (4.27) we have droppedpossible Majorana phases. One can use the recent global analysis of data to determine thevarious entries of U [32]. We have taken the phase factor δ to be zero for simplicity. Then,using the central values of all angles, including that for θ13 as obtained from the recentDaya Bay and RENO experiments [29, 30], the right-hand side of equation (4.25) is com-pletely known, at least in orders of magnitude. The actual mass matrix thus constructedhas some elements at least one order of magnitude smaller than the others, thus suggest-ing texture zeros.

For each of the benchmark points used in the next section, w1 and w2, the vevs of thetwo triplets, are determined by values of the parameters in the scalar potential. Of course,the coupling matrices h(1) and h(2) are still indeterminate. In order to evolve a workingprinciple based on economy of free parameters, we fix the Yukawa coupling matrix h(2) bychoosing one suitable value for all elements of the µ–τ block and another value, a smallerone, for the rest of the matrix. That fixes all the elements of the other matrix. Althoughthere is a degree of arbitrariness in such a method, we emphasize that it does not affectthe generality of our conclusions, so long as we adhere to the wide choice of scenariosadopted in the next section, including both small and large values of the ∆L = 2 Yukawacouplings.

4.4 Benchmark points and doubly-charged scalar decays

Our purpose is to investigate the expected changes in the phenomenology of doubly-charged scalars when two triplets are present. In general, the two scalars of this kind,

57


namely, H++1 and H++

2 can both be produced at the LHC via the Drell-Yan process, whichcan have about 10% enhancement from the two-photon channel. They will, over a largeregion of the parameter space, have the following decays [156]:

H++1 → `+

i `+j , (4.29)

H++1 → W+W+, (4.30)

H++1 → H+

2 W+, (4.31)

H++2 → `+

i `+j , (4.32)

H++2 → W+W+, (4.33)

with `i, `j = e, µ in equation (4.29). As we discussed in section 4.2, in the context of thesingle-triplet model the decay analogous to equation (4.31) is practically never allowed,unless the masses are very low. On the other hand, mixing between two triplets opensup situations where the mass separation between H++

1 and H+2 kinematically allows the

transition (4.31). Denoting the mass of H++k by Mk and that of H+

k by µk (k = 1, 2) andusing the convention M1 > M2 and µ1 > µ2, this decay is possible if M1 > µ2 + mW ,where mW is the mass of W boson. We demonstrate numerically that this can naturallyhappen, by considering three distinct regions of the parameter space and selecting fourbenchmark points (BPs) for each region.

We have seen that, in a model with a single triplet, the doubly-charged Higgs decaysinto either `+

i `+j or W+W+. The former is controlled by the ∆L = 2 Yukawa coupling con-

stants hij , while the latter is driven by the triplet vev w. Since neutrino masses are givenby Mν = hw, large (' 1) values of hij imply a small vev w, and vice versa. Accordingly,assuming hij 6= 0, three regions in the parameter space can be identified, where one canhave [156]

i) Γ(∆++ → `+i `

+j ) Γ(∆++ → W+W+),

ii) Γ(∆++ → `+i `

+j ) Γ(∆++ → W+W+),

iii) Γ(∆++ → `+i `

+j ) ∼ Γ(∆++ → W+W+).

In the context of two triplets, we choose three different ‘scenarios’ in the same spirit, withsimilar relative rates of the two channels H++

k → `+i `

+j and H++

k → W+W+. Four BPsare selected for each such scenario through the appropriate choice of parameters in thescalar potential. The resulting masses of the various physical scalar states are shown intables 4.1 and 4.2. Although our study focuses mainly on the phenomenology of charged

58

4.5. USEFULNESS OF H++1 −→ H+

2 W+ AT THE LHC

scalars, we also show the masses of the neutral scalars. It should be noted that the lightestCP-even neutral scalar, which is dominated by the doublet, has mass around 125 GeV foreach BP. It has also been checked that, here as well as in the situation discussed in thenext chapter, the benchmark points used by us are consistent with the observed signalstrengths of the 125 GeV scalar, within 1σ level [158].

All the twelve BPs (distributed among the three different scenarios) have M1 suffi-ciently above µ2 to open upH++

1 → H+2 W

+. The branching ratios in different channels areof course dependent on the specific BP. We list all the branching ratios forH++

1 andH++2 in

table 4.3, together with their pair-production cross sections at the LHC with√s = 14 TeV.

The cross sections and branching ratios have been calculated with the help of the pack-age FeynRules (version 1.6.0) [159,160], thus creating a new model file in CompHEP (ver-sion 2.5.4) [161]. CTEQ6L parton distribution functions have been used, with the renor-malisation and factorisation scales set at the doubly-charged scalar mass. Using the fullmachinery of scalar mixing in this model, the decay widths into various channels havebeen obtained.

The results summarised in table 4.3 show that, for the decay of H++1 , the channel

H+2 W

+ is dominant for two of the four BPs in scenario 1 and all four BPs in scenar-ios 2 and 3. This, in the first place, substantiates our claim that one may have to lookfor a singly-charged scalar in the final state that opens up when more than one doubletis present. This is because, for the BPs where H++

1 → H+2 W

+ dominates, the branchingratios for the other final states are far too small to yield any detectable rates.

4.5 Usefulness of H++1 −→ H+

2 W+ at the LHC

Table 4.3 contains the rates for pair-production of the heavier as well as the lighter doubly-charged scalar at the 14 TeV run of the LHC. A quick look at these rates revals that forthe heavier of the doubly-charged scalars, it varies from about 1.4 fb to 3.6 fb, for massesranging approximately between 400 and 550 GeV. Therefore, as can be read off from ta-ble 4.3, for ten of our twelve BPs, an integrated luminosity of about 500 fb−1 is likely toyield about 700 to 1800 events of the H+

2 W+H−2 W

− type. Keeping in mind the fact thatH+

2 mostly decays in the channel H+2 → `+ν`, such final states should prima facie be ob-

served at the LHC, although event selection strategies of a very special nature may berequired to distinguish the H+

2 from a W+ decaying into `+ν`.The primary advantage of focusing on the channelH++

1 −→ H+2 W

+ is that it helps onein differentiating between the two kinds of type II cases, namely those containing one and

59


two scalar triplets, respectively. In order to emphasize this point, we summarize belowthe result of a simulation in the context of the 14 TeV run of the LHC. For our simulation,the amplitudes have been computed using the package Feynrules (version 1.6.0), withthe subsequent event generation through MadGraph (version 5.12) [162], and showeringwith the help of PYTHIA 8.0. CTEQ6L parton distribution functions have been used.

We compare the two-triplet case with the single-triplet case. In the first case, there aretwo doubly charged scalars, and one has contributions from both H±±1 and H±±2 to theleptonic final states following their Drell-Yan production. While the former, in the chosenbenchmark points, decays into H±1 W±, the latter goes either to a same-sign W -pair or tosame-sign dileptons. If one considers two, three and four-lepton final states with missingtransverse energy (MET), there will be contributions from both of the doubly-chargedscalars, with appropriate branching ratios, combinatoric factors and response to the cutsimposed. We have carried out our analysis with a set of cuts listed in table 4.4, whichare helpful in suppressing the standard model backgrounds. Thus one can define thefollowing ratios of events emerging after the application of cuts [156]:

r1 =σ(4`+ MET)

σ(3`+ MET), r2 =

σ(4`+ MET)

σ(2`+ MET). (4.34)

The values of these ratios for the three scenarios of BP 3 are presented in table 4.5.In each case, the ratios for the two-triplet case is presented alongside the correspondingsingle-triplet case, with the mass of the doubly charged scalar in the latter case beingclose to that of the lighter state H±±2 in the former case. Both of the situations where, inthe later case, the doubly charged scalar decays dominantly into either W±W± or `±`±

are represented in our illustrative results. One can clearly notice from the results (whichapply largely to our other benchmark points as well) that both r1 and r2 remain substan-tially larger in the two-triplet case as compared to the single-triplet case. One reason forthis is an enhancement via the combinatoric factors in the two-triplet case. However,the more important reason is that the 4` events survive the missing transverse energy(MET) cut with greater efficiency. In the single-triplet case, the survival rate efficiencyis extremely small when H±± decays mainly into same-sign dileptons, the MET comingmostly from energy-momentum mismeasurement (as a result of lepton energy smearing)or initial and final-state radiation. In the two-triplet case, on the other hand, the decayH++

1 −→ H+2 W

+ leaves ample scope for having MET in W -decays as well as in the decayH+

2 −→ `+ν`, thus leading to substantially higher cut survival efficiency. Thus, from anexamination of such numbers as those presented in table 4.5, one can quite effectively use

60


2 W+ AT THE LHC

Mass (GeV) BP 1 BP 2 BP 3 BP 4

H++1 515.99 515.99 521.54 524.15

Scenario 1 H++2 443.04 429.16 455.59 470.15

H+1 515.98 515.98 498.97 515.78

H+2 368.45 360.15 423.26 418.65

H++1 526.78 525.00 429.13 464.31

Scenario 2 H++2 414.18 401.63 392.45 407.20

H+1 520.26 519.86 414.48 459.23

H+2 343.28 334.97 339.02 340.63

H++1 521.54 464.31 525.00 429.13

Scenario 3 H++2 455.59 407.20 401.63 392.45

H+1 498.97 459.23 519.86 414.48

H+2 423.26 340.63 334.97 339.02

Table 4.1: Charged scalar masses.

the channel H++1 −→ H+

2 W+ to distinguish a two-triplet case from a single-triplet case,

provided the heavier doubly-charged state is within the kinematic reach of the LHC.

61


Mass (GeV) BP 1 BP 2 BP 3 BP 4

H01R 365.70 364.86 350.39 364.59

H02R 193.89 194.00 256.09 245.96

Scenario 1 H03R 125.00 125.03 125.01 125.01

H01I 364.98 364.85 350.39 364.59

H02I 194.43 193.98 256.08 245.96

H01R 365.69 365.70 295.58 325.51

H02R 173.97 173.96 173.98 173.96

Scenario 2 H03R 125.02 125.02 125.04 125.02

H01I 365.69 365.70 295.59 325.52

H02I 173.97 173.96 173.98 173.96

H01R 350.39 325.51 365.69 295.58

H02R 256.08 173.96 173.98 173.96

Scenario 3 H03R 125.02 125.02 125.04 125.02

H01I 350.39 325.51 365.69 295.58

H02I 256.08 173.96 173.98 173.96

Table 4.2: Neutral scalar masses.

62


2 W+ AT THE LHC

Data BP 1 BP 2 BP 3 BP 4

BR(H++1 → H+

2 W+) 0.08 0.10 0.99 0.99

BR(H++1 → W+W+) 0.92 0.90 0.01 0.004

BR(H++1 → `+

i `+j ) < 10−16 < 10−16 < 10−19 < 10−20

Scenario 1 BR(H++2 → W+W+) 0.99 0.99 0.99 0.99

BR(H++2 → `+

i `+j ) < 10−19 < 10−19 < 10−17 < 10−18

σ(pp→ H++1 H−−1 ) 1.664 fb 1.534 fb 1.446 fb 1.408 fb

σ(pp→ H++2 H−−2 ) 3.044 fb 3.5 fb 2.714 fb 2.308 fb

BR(H++1 → H+

2 W+) 0.99 0.99 0.99 0.98

BR(H++1 → W+W+) < 10−21 < 10−21 < 10−17 < 10−20

BR(H++1 → `+

i `+j ) 0.01 0.01 0.001 0.02

Scenario 2 BR(H++2 → W+W+) < 10−18 < 10−18 < 10−14 < 10−18

BR(H++2 → `+

i `+j ) 0.99 0.99 0.99 0.99

σ(pp→ H++1 H−−1 ) 1.36 fb 1.41 fb 3.59 fb 2.46 fb

σ(pp→ H++2 H−−2 ) 3.98 fb 4.65 fb 5.28 fb 4.38 fb

BR(H++1 → H+

2 W+) 0.99 0.99 0.99 0.99

BR(H++1 → W+W+) < 10−12 < 10−10 < 10−11 < 10−9

BR(H++1 → `+

i `+j ) < 10−9 < 10−11 10−11 < 10−11

Scenario 3 BR(H++2 → W+W+) 0.0001 0.98 0.97 0.99

BR(H++2 → `+

i `+j ) 0.99 0.02 0.03 0.01

σ(pp→ H++1 H−−1 ) 1.45 fb 2.46 fb 1.41 fb 3.59 fb

σ(pp→ H++2 H−−2 ) 2.71 fb 4.38 fb 4.65 fb 5.28 fb

Table 4.3: Decay branching ratios and production cross sections for doubly-chargedscalars.

63


MET > 70 GeV

Σ|pvisT |+ MET > 500 GeV∣∣∣plepton

T

∣∣∣ > 30 GeV

|η lep| < 2.5

|η jet| < 4.5

Table 4.4: Cuts used for determination of ratios of events r1 and r2. The subscript T standsfor ‘transverse’ and η denotes the pseudorapidity.

BP 3 Ratio Two triplets One triplet

Scenario 1r1 0.20 0.04

r2 0.05 0.01

Scenario 2r1 0.44 < 10−6

r2 0.21 < 10−9

Scenario 3r1 0.12 < 10−5

r2 0.04 < 10−6

Table 4.5: Ratio of events r1, r2 for two-triplet and single-triplet scenario respectively forbenchmark point 3.

4.6 Conclusions

In this chapter, we have argued, taking models with the type II seesaw mechanism forneutrino mass generation as a motivation, that it makes sense to consider scenarios withmore than one scalar triplet. As the simplest extension, we have formulated in detail amodel with two Y = 2 complex triplets of this kind. On taking into account the mix-ing of the triplets with each other (and also with the doublet, albeit with considerablerestriction), and thus identifying all the mass eigenstates along with their various interac-tion strengths, we find that the heavier doubly-charged scalar decays dominantly into thelighter singly-charged scalar and a W boson over a large region of the parameter space. Itshould be re-iterated that this feature is a generic one and is avoided only in very limitedsituations or in the case of unusually high values of the triplet Yukawa coupling. Thedeciding factor here is the decay being driven by the SU(2) gauge coupling.

Thus the above mode is often the only way of looking for the heavier doubly-chargedscalar state and thus for the existence of two scalar triplets. Our choice of benchmarkpoints for reaching this conclusion spans cases where the ∆L = 2 lepton couplings of

64

4.6. CONCLUSIONS

the triplets have values at the high (close to one) and low as well as the intermediatelevel, consistent with the observed neutrino mass and mixing patterns. In general, withthe heavier triplet mass ranging up to more than 500 GeV, one expects about 700 to 1800events of the type pp → H+

2 W+H−2 W

− at the 14 TeV run of the LHC, for an integratedluminosity of 500 fb−1. We have also demonstrated that ratios of the numbers of two,three and four-lepton events with MET offer a rather spectacular distinction of the two-triplet case from one with a single triplet only.

For unsuppressed production of doubly charged scalars, the Drell - Yan pair produc-tion mode is useful, since no suppression by triplet vev’s comes into play. On the otherhand, the smallness of the triplet vev also makes the single production rate (via gauge bo-son fusion) of doubly charged scalar rather small [163]. Of course, for H++H−− produc-tion, there are possibilities of detecting a doubly charged Higgs at the colliders throughthe decays H++ → l+l+ and H++ → W+W+. In these cases it is interesting to notethat the interplay of the two parameters, the triplet Yukawa coupling hll and triplet vevw actually govern the relative strengths of the two decays channels H++ → l+l+ andH++ → W+W+, as for the identification of the H±± signal. We would like to mentionthat the doubly charged scalars may be detected at the LHC for the l±l± decay mode with300fb−1 integrated luminosity, upto a mass range of 1TeV even with a branching fraction∼ 60% into this channel. But, in spite of the distinctive nature of the signal produced byH++ → l+l+ decay, their faking by a number of sources cannot be ruled out [163]. On theother hand, in the complementary region of parameter space, doubly charged scalars aremost likely to be observed upto a mass of 700GeV in the H++ → W+W+ decay mode.But, it is worth mentioning that signals from this decay mode are in general more back-ground prone than the l+l+ decay mode [163]. The techniques used in [163] for the decayH++ → W+W+ can be extended to the decay H++ → H+W+ for background suppres-sion.

65


66

Chapter 5

CP-violating phase in a two Higgs tripletscenario : some phenomenologicalimplications

5.1 Introduction

We begin by recapitulating the theme explored in the last chapter. We have already seenthat type-II seesaw models can generate Majorana neutrino masses without any right-handed neutrino(s) with the help of one or more Y=2 scalar triplets. The restriction ofthe vacuum expectation value (vev) of such a triplet, arising from the limits on the ρ-parameter (with ρ = m2

W/m2Z cos2 θ), is obeyed in a not so unnatural manner, where the

constitution of the scalar potential can accommodate large triplet scalar masses vis-a-vis asmall vev. In fact, this very feature earns such models classification as a type of ‘seesaw’.

A lot of work has been done on the phenomenology of scalar triplets which, interest-ingly, also arise in left-right symmetric theories [164]. One can, however, still ask the ques-tion: is a single-triplet scenario self-sufficient, or does the replication of triplets (togetherwith the single scalar doublet of the SM) bring about any difference in phenomenology?This question, otherwise a purely academic one, acquires special meaning in the contextof some neutrino mass models which aims to connect the mass ordering with the val-ues of the mixing angles, thereby achieving some additional predictiveness. We havementioned in the previous chapters that a class of such models depend on texture zeros,where a number of zero entries (usually restricted to two) in the mass matrix enable oneto establish the desired connection.

The existence of such zero entries require the imposition of some additional symmetry.We have pointed out that the simultaneous requirement of zero textures and the type-II

67

CHAPTER 5. CP-VIOLATING PHASE IN A TWO HIGGS TRIPLET SCENARIO : SOMEPHENOMENOLOGICAL IMPLICATIONS

seesaw mechanism,however, turns out to be inconsistent. The inconsistency is gone fortwo or more triplets. This resurrects the relevance of the phenomenology of two-triplet,one doublet scalar sectors, this time with practical implications. We have studied suchphenomenology in the previous chapter. An important conclusion of that study was that,whereas the doubly charged scalar in the single triplet scenario would decay mostly inthe `±`± or W±W± modes, the decay channel H++

1 −→ H+2 W

+ acquires primacy over alarge region of the parameter space. Some predictions on this in the context of the LargeHadron Collider were also shown. However, an added possibility with two triplet is thepossibility of at least one CP-violating phase being there. This in principle can affect thephenomenology of the model, which is worth studying.

With this in view, we have analyzed in the present chaper the one-doublet, two-triplet framework, including CP-violating effects arising via a relative phase between thetriplets. Thus, the vev of one triplet has been made complex and consequently, the coef-ficient of the corresponding trilinear term in the scalar potential has also been renderedcomplex.

Indeed, the introduction of a phase results in some interesting findings that were notpresent when the relative phase was absent. First of all, as a result of mixing betweentwo triplets and presence of a relative phase between them, something on which no phe-nomenological restrictions exits, the heavier doubly charged scalar can dominantly decayinto the lighter doubly charged scalar plus the SM-like Higgs boson i.e. H++

1 → H++2 h,

over a larger range of parameter space. This can give rise to a spectacular signal in thecontext of LHC. Basically, as a final state, we obtain H++

1 → `+`+h i.e a doubly chargedscalar decaying into two same-sign leptons plus the SM-like Higgs. This decay oftendominates over all other decay channels. When this decay is not present due to insuf-ficient mass-difference between respective scalars, the decay H++

2 −→ H+2 W

+ mostlydominates, and its consequence was discussed in some detail in the previous chapter onthe CP-conserving scenario.

Secondly, for some combination of parameters, the decays mentioned in the aboveparagraph are not possible with a vanishing or small phase, due to insufficient gap inmasses between the respective scalars. However, if we continuously increase the valueof the phase, keeping all the other parameters fixed, the mass differences between thescalars start to increase, so that the aforementioned channel finally opens up.

Thirdly, we have noticed in the previous chapter that the gauge coupling domi-nated decay H++

1 → H+2 W

+ dominates over the Yukawa coupling dominated decay∆++ → `+`+, even in those region of parameter space where we have chosen the Yukawa

68

5.2. A SINGLE SCALAR TRIPLET WITH A CP-VIOLATING PHASE

coupling matrices to be sufficiently large (' 1). On the other hand, the CP-violating phasesuppresses the neutrino mass matrix elements for the same value of the triplet vacuumexpectation values (vev). This in turn requires an increase in the corresponding Yukawacoupling matrix elements, since the vev’s and Yukawa couplings are related by the ex-pression for neutrino masses. The outcome of this whole process is that, for several BPs,the decay H++

1 → `+`+ competes with the decay into H+2 W

+.

Finally, the CP-conserving scenario marks out regions of the parameter space, wherethe branching ratios of the decays H++

1 −→ `+`+ and H++1 → W+W+ are of compara-

ble, though subdominant, rates. As the phase picks up, the same vev causes a hike inYukawa coupling, as discussed above. In such situations, the decay H++

1 → `+`+ mostlydominates over the W+W+ mode.

Since, to the best of our knowledge, very little has been written on CP-violatingphase(s) in two-triplet scenarios, we introduce the issue in a minimalistic manner, in-troducing one such phase. The presence of additional phases can of course subject theallowed parameter space to new possibilities. We postpone their discussion for a furtherstudy.

We begin by presenting a summary of the scenario with a single triplet with complexvev in the next section.

5.2 A single scalar triplet with a CP-violating phase

We first give a glimpse of the scenario with a single triplet ∆ = (∆++,∆+,∆0), over andabove the usual Higgs doublet φ, using the notation of [155]. ∆ is equivalently denotedby the 2× 2 matrix

∆ =

(∆+

√2∆++

√2∆0 −∆+

). (5.1)

The vev’s of the doublet and the triplet are expressed as

〈φ〉0 =1√2

(0

v

)and 〈∆〉0 =

(0 0

vT 0

), (5.2)

respectively. The only doublet-dominated physical state that survives after the generationof gauge boson masses is a neutral scalar h.

69


The most general scalar potential including φ and ∆ can be written as [155]

V (φ,∆) = a φ†φ+b

2Tr (∆†∆) + c (φ†φ)2 +

d

4

(Tr (∆†∆)

)2

+e− h

2φ†φTr (∆†∆) +

f

4Tr (∆†∆†) Tr (∆∆)

+ hφ†∆†∆φ+(t φ†∆φ+ h.c.

), (5.3)

where φ ≡ iτ2φ∗ and τ2 is the Pauli matrix. All parameters in the Higgs potential are

real except t which is complex in general. By performing a global U(1) transformation, vcan always be chosen real and positive. Because of the t-term in the potential there is nosecond global symmetry to make vT real. As already mentioned, t can also be complexand, therefore, it can be written as t = |t| eiα and vT = weiγ with w ≡ |vT |. Minimizationof the scalar potential with respect to the phase of vT i.e γ, gives the relation between thephases as α + γ = π. 1

The choice a < 0, b > 0 ensures that the dominant source of spontaneous symmetrybreaking is the scalar doublet. It is further assumed, following [155], that

a, b ∼ v2; c, d, e, f, h ∼ 1; |t| v. (5.4)

Such a choice is motivated by the following considerations as mentioned in the previouschapter

• The need to fulfill the electroweak symmetry breaking conditions,

• To have w v sufficiently small, as required by the ρ-parameter constraint,

• To keep doublet-triplet mixing low in general, and

• To ensure that all quartic couplings are perturbative.

The mass terms for the singly-charged scalars can be expressed in a compact form as

L±S = −(H−, φ−)M2+

(H+

φ+

)(5.5)

with

M2+ =

((q + h/2)v2

√2v(t∗ − vTh/2)

√2v(t− v∗Th/2) 2(q + h/2)w2

)and q =

|t|w. (5.6)

1For an analogous situation with two Higgs doublets, see, for example [165].

70

5.3. TWO SCALAR TRIPLETS AND A CP-VIOLATING PHASE

Keeping aside the charged Goldstone boson, the mass-squared of the singly-chargedphysical scalar is obtained as

m2H+ =

(q +

h

2

)(v2 + 2w2), (5.7)

while the doubly-charged scalar mass is expressed as

m2∆++ = (h+ q)v2 + 2fw2. (5.8)

Thus, in the limit w v,

m2∆++ −m2

H+ 'h

2v2. (5.9)

Therefore, a substantial mass splitting between ∆++ andH+ is in general difficult. Thistends to disfavour the H+W+ decay channel of ∆++, as compared to `+`+ and W+W+, asalready been pointed out in the previous chapter.

5.3 Two scalar triplets and a CP-violating phase

We write down the expressions for the triplets and the scalar potential involving themonce more for the sake of completeness. The Y = 2 triplet scalars ∆1, ∆2, both written as2×2 matrices [147]:

∆1 =

(δ+

1

√2δ++

1√2δ0

1 −δ+1

)and ∆2 =

(δ+

2

√2δ++

2√2δ0

2 −δ+2

). (5.10)

The vev’s of the scalar triplets are given by [147]

〈∆1〉0 =

(0 0

w1 0

)and 〈∆2〉0 =

(0 0

w2 0

). (5.11)

The vev of the Higgs doublet is as usual given by equation (5.2).The scalar potential in this model involving φ, ∆1 and ∆2 is given by [147],

V (φ,∆1,∆2) =

a φ†φ+1

2bkl Tr (∆†k∆l) + c(φ†φ)2 +

1

4dkl

(Tr (∆†k∆l)

)2

+1

2(ekl − hkl)φ†φTr (∆†k∆l) +

1

4fkl Tr (∆†k∆

†l ) Tr (∆k∆l)

+hkl φ†∆†k∆lφ+ g Tr (∆†1∆2) Tr (∆†2∆1) + g′ Tr (∆†1∆1) Tr (∆†2∆2)

+(tk φ

†∆kφ+ H.c.), (5.12)

71


where summation over k, l = 1, 2 is understood.In the previous chapter, all the vev’s as well the parameters in the potential were as-

sumed to be real. As has already been mentioned, this need not be the situation in general.To see the phenomenology including CP-violation, we make a minimal extension of thesimplified scenario by postulating one CP-violating phase to exist. This entails a complexvev for any one triplet (in our case we have chosen it to be ∆1). At the same time, thereis a complex phase in the coefficient t1 of the trilinear term in the potential. Thus one canwrite t1 = |t1| eiβ and w1 = |w1| eiα.

Using considerations very similar to those for the single-triplet model, we have taken

a, bkl ∼ v2; c, dkl, ekl, hkl, fkl, g, g′ ∼ 1; |tk| v. (5.13)

We have chosen to restrict ourselves to cases where w1, w2 v, keeping in mind theconstraint on the ρ-parameter.

As in the previous chapter, we use the smallness of the triplet vev’s wk, and drop thequartic terms in the scalar triplets during the diagonalisation of the mass matrices. Thisenables one to use approximate analytical expressions, which makes our broad conclu-sions somewhat transparent. However, the numerical results presented in section 5.4 areobtained using the full potential (5.12), including the effects of the triplet vev’s.

For convenience we define the following matrices and vectors [147]:

B = (bkl), E = (ekl), H = (hkl), (5.14)

t =

(|t1|cosβt2

), t′ =

(|t1|sinβ

0

), w =

(|w1|cosαw2

), w′ =

(|w1|sinα

0

). (5.15)

In terms of the parameters defined in equations 5.14 and 5.15, the conditions for a station-ary point of the potential are [147] (

B +v2

2(E −H)

)w + v2 t = 0, (5.16)

a+ cv2 +1

2wT (E −H)w + 2 t · w + 2t′ · w′ + 1

2w′T (E −H)w′ = 0, (5.17)

(b11 +v2

2(e11 − h11))|w1|sinα− v2|t1|sinβ = 0, (5.18)

using the notation t · w =∑

k tkwk. These three equations are exact if one neglects allterms quartic in the triplet vev’s in V0 ≡ V (〈φ〉0, 〈∆〉0). In equation (5.17) we have already

72


divided by v, assuming v 6= 0. The small vev’s wk are thereafter obtained as [147]

w = −v2

(B +

1

2v2(E −H)

)−1

t. (5.19)

And from equation (5.18) the phase of t1 can be expressed as [147]

sinβ =v−2(b11 + v2

2(e11 − h11))|w1|sinα|t1|

(5.20)

This re-iterates the fact that the phases t1 and w1 are related to each other. It is also evidentfrom (5.20) that the value of the angle α has to be nπ where n = 0, 1, 2, 3....when the phaseβ is absent.

Next we discuss the mass matrices of charged scalars. The mass matrix of the doubly-charged scalars is obtained as [147]

M2++ = B +

v2

2(E +H) . (5.21)

It is interesting to note that if we drop those quartic terms for simplification from ourscalar potential, then our doubly charged mass matrix remains the same as in the previouschapter. This gives the impression that the relative phase between triplets does not affectthe doubly charged mass matrix if we drop the quartic terms in the potential. But in ournumerical calculation, where we have taken the full scalar potential including the quarticterms, we find such a dependence, arising obviously from the quartic terms. This will bediscussed further in the next section.

As for the singly-charged fields ∆+k , one has to consider their mixing with φ+, the

would-be Goldstone boson of the Standard Model. We write the mass term as [147]

−L±S =(δ−1 , δ

−2 , φ

−)M2+

δ+1

δ+2

φ+

+ h.c., (5.22)

equation (5.12) leads to [147]

M2+ =

(B + v2

2E

√2v (t−Hw/2)√

2v (t−Hw/2)† a+ cv2 + 12wT (E +H)w + 1

2w′T (E +H)w′

). (5.23)

This mass matrix must have a zero eigenvalue, corresponding to the Goldstone boson.Indeed, on substituting the minimization equations (5.16), (5.17) and (5.18), we findthat [147]

Det(M2+) = 0, (5.24)

73


which ensures a consistency check.As in the previous chapter, the mass matrices (5.21) and (5.23) are diagonalized

by [147]U †1M2

++U1 = diag (M21 ,M

22 ) and U †2M2

+U2 = diag (µ21, µ

22, 0), (5.25)

respectively, with [147]

(δ++

1

δ++2

)= U1

(H++

1

H++2

),

δ+1

δ+2

φ+

= U2

H+1

H+2

G+

. (5.26)

We have denoted the fields with definite mass by H++k and H+

k ; and G+ is the chargedGoldstone boson.

We also outline the neutral sector of the model, which now cannot be separated intoCP-even and CP-odd sectors. Thus the mass matrix for the neutral sector of the presentscenario turns out to be a 6 × 6 matrix , including mixing between real and imaginaryparts of the complex neutral fields. The symmetric neutral mass matrix is denoted byMneut. So, the mass term for the neutral part can be written as [147] :

−L0S = (N01, N02, N03, N04, N05, N06)M2

neut

N01

N02

N03

N04

N05

N06

+ h.c., (5.27)

Where N0n’s are the neutral states in flavor basis. This mass matrix is diagonalizedby [147]

U †3M2neutU3 = diag (M2

01,M202,M

203,M

204,M

2h , 0) (5.28)

with [147]

N01

N02

N03

N04

N05

N06

= U3

H01

H02

H03

H04

h

G0

. (5.29)

Where h is identified with the SM-like Higgs boson andG0 is the neutral Goldstone boson.

74


It is interesting to note that once we remove the phases of coefficient of trilinear termin scalar potential and the vev of triplet H++

1 by setting α = β = 0 , then the mixingbetween the CP-even and CP-odd scalars vanishes and we get back the usual separate3× 3 matrices for these two sectors. This also serves as a consistency check for the model.

The ∆L = 2 Yukawa interactions of the triplets are [147]

LY =1

2

2∑k=1

y(k)ij L

Ti C−1iτ2∆kLj + h.c., (5.30)

where C is the charge conjugation matrix, the y(k)ij are the symmetric Yukawa coupling

matrices of the triplets ∆k, and the i, j are the summation indices over the three neutrinoflavours. The charged-lepton mass matrix is diagonal in this basis.

The neutrino mass matrix is generated from LY as [147]

(Mν)ij = y(1)ij |w1|cosα + y

(2)ij w2. (5.31)

This relates the Yukawa coupling constants y(1)ij , y(2)

ij and the real part of the triplet vev’s,namely, |w1|cosα and w2.

The neutrino mass eigenvalues are fixed according to a particular type of mass spec-trum. In this chapter also, we illustrate our points, without any loss of generality, in thecontext of normal hierarchy, setting the lowest neutrino mass eigenvalue to zero. Again,the elements of the neutrino mass matrix Mν can be found by using

Mν = U †MνU, (5.32)

where U is the PMNS matrix already defined in the previous chapter. Mν is the diagonalmatrix of the neutrino masses. We have followed the same procedure, as described in theprevious chapter, to compute the elements of U . Also, the phase δ has been set to zero.For θ13, we have used the results from the Daya Bay and RENO experiments [29, 30].

All terms on the right-hand side of equation (5.30) are approximately known, whichis sufficient for predicting phenomenology in the 100 GeV - 1 TeV scale.. The actual massmatrix thus constructed, on numerical evaluation, approximately reflects a two-zero tex-ture which is one of the motivations of this study.

For each benchmark point used in the next section,w1 andw2 get determined by valuesof the other parameters in the scalar potential. Of course, the coupling matrices y(1) andy(2) are still indeterminate in this case also. We fix the matrix elements of y(2) and y(1) fol-lowing the same method used in the previous chapter and it has already been mentionedthere that our broad conclusions do not depend on this ‘working rule’.

75


Of couse, the success of a two-triplet scenario in the context of a seesaw mechanism re-quires the electroweak vacuum to be (meta)stable till at least the seesaw scale. In general,the tendency of the top quark yukawa coupling to turn the doublet quartic couplings neg-ative is resposible for the loss of stability. Additional scalar quartic couplings usually off-sets this effect [166], and the present scenario is no exception. It been shown in [167] thatone can ensure stability upto 1016−18GeV with a single triplet. With the low-energy quar-tic couplings not too different from these and with one more triplet interacting with thedoublet, the situation is even more optimistic. Moreover, the acceptence of a metastableelectroweak vacuum can help the scenario even further.

5.4 Benchmark points and numerical predictions

The trademark signal of Higgs triplets is contained in the doubly charged components. Inthe current scenario, too, one would like to see the signatures of the two doubly chargedscalars, especially the heavier one, namely H++

1 whose decays have already been shownto contain a rather rich phenomenology.

The H++1 , produced at the LHC via the Drell-Yan process can, in general, have two-

body decays through the following channels [147]:

H++1 → H++

2 h, (5.33)

H++1 → `+

i `+j , (5.34)

H++1 → W+W+, (5.35)

H++1 → H+

2 W+, (5.36)

H++2 → `+

i `+j , (5.37)

H++2 → W+W+, (5.38)

with h is the SM-like Higgs and `i, `j = e, µ.The decay modes (5.33)and (5.36) are absent in the single-triplet model. On the other

hand, mixing between two triplets opens up situations where the mass separation be-tween H++

1 , H++2 and H++

1 , H+2 is sufficient to kinematically allow the transitions (5.33)

and (5.36). The decay (5.33) opens up a spectacular signal, especially when H++2 mostly

decays into two same sign leptons, leading to

H++1 → `+

i `+j h (5.39)

76

5.4. BENCHMARK POINTS AND NUMERICAL PREDICTIONS

Let us denote the mass of SM Higgs by Mh, that of H++k by Mk and that of H+

k by µk

(k = 1, 2). Then, in the convention M1 > M2, µ1 > µ2, the decays (5.33)and (5.36) arepossible only if M1 > M2 + Mh and M1 > µ2 + mW . We demonstrate numerically thatthis can naturally happen by considering three distinct regions of the parameter spaceand selecting four benchmark points (BPs) for each region. The relative phase betweentwo triplets also a plays an important roll in these cases. In order to emphasise this, wehave also chosen three different values of the phase, namely α = 30, 45 and 60 for eachbenchmark point. Thus we have considered 36 BPs alltogether, comprising three distinctregions of the parameter space and relative phases between triplets to justify our findings.

We have seen that, in a single-triplet model, the doubly-charged Higgs decays into ei-ther `+

i `+j or W+W+. The former is controlled by the ∆L = 2 Yukawa couplings yij , while

the latter is driven by w, the triplet vev. Neutrino masses are given by (5.31), implyinglarge values of yij for small w and vice versa. Interestingly, the presence of triplet phasethrough the cosα term in this equation actually suppresses the effect of vev w1 of the firsttriplet. This in turn implies that we get higher values for Yukawa coupling matrix en-tries yij1 compared to the case where CP-violating effects are absent. In this chapter also,we have identified, for the chosen values of triplet phase, three regions in the parameterspace, corresponding to [147]

i) Γ(H++1,2 → `+

i `+j ) Γ(H++

1,2 → W+W+),

ii) Γ(H++1,2 → `+

i `+j ) Γ(H++

1,2 → W+W+),

iii) Γ(H++1,2 → `+

i `+j ) ∼ Γ(H++

1,2 → W+W+).

These are referred to as scenarios 1, 2 and 3 respectively in the subsequent discussion.The masses of the various physical scalars and some of their phenomenological prop-

erties are shown in Tables 5.1-5.9. Although our study involves mainly the phenomenol-ogy of charged scalars, we have also listed the masses of neutral scalars. Values of theparameters occuring in the scalar potential have been used appropriately for obtainingthese benchmark values of masses. Once more, there emerges a 125 GeV scalar whoseinteractions are consistent with the observed signal strengths at the LHC.

In the previous chapter, we had concentrated on those benchmark points in the pa-rameter space, for whichH++

1 → H+2 W

+ becomes a dominant decay mode. In the presentchapter, we draw the reader’s attention to an interesting complementary situation : onecan have, in certain regions of the parameter space, H++

1 → H++2 h as the dominant chan-

nel in H++1 decay. Since the triplet masses are free parameters, this can of course happen

77


without any ’theoretical design’. However. the presence of the CP-violating phase canalso play an interesting role here. This is demonstated in Figure 5.1. To understand thesituation, suppose the mass parameters in the potential are fixed in such a way that thedecay H++

1 → H++2 h is not possible for α = 0. Now, if the CP-violating phase α is grad-

ually increased from zero, keeping all other parameters fixed, then both the mass differ-ences mH++

1−mH++

2and mH++

1−mH+

2start increasing rather sharply for α & 60. This is

because the degree of doublet-triplet mixing for ∆1 in our parametrisation changes withα. In such situations, as shown, for example, in Table 5.9, the quartic couplings causethe decay H++

1 → H++2 h to dominate over H++

1 → H+2 W

+. Thus, other than the free pa-rameters corresponding to the scalar masses, the CP-violating phase has a part to play inthe phenomenology of a two-triplet scenario. One consequence of this will be discussedbelow.

Earlier, we neglected contributions from the quartic terms in our scalar potential inthe approximate forms of the doubly and singly-charged mass matrices. However, theimport of the phase is not properly captured unless one retains these terms. Thus it isonly via a full numerical analysis of the potential retaining all terms that the above effectof the phase of the trilinear term becomes apparent.

It should also be noted that the cosine of the complex phase suppresses the contri-bution to neutrino masses. Consequently, for the same triplet vev, one requires largervalues of the Yukawa interaction strengths. This makes the l+l+ decay mode of a doublycharged scalar more competitive with W+W+, as compared to the results presented inthe previous chapter.

The branching ratios for a given scalar in different channels are of course dependenton the various parameters that characterise a BP. We list all the charged scalar massesin Tables 5.1, 5.4 and 5.7. Moreover, the neutral scalar masses are shown in Tables 5.2,5.5 and 5.8 for three different values of triplet phase α. The branching ratios for H++

1

and H++2 for different triplet phases are listed in Tables 5.3,5.6 and 5.9, together with

their pair-production cross sections at the LHC with√s = 13 TeV. The cross sections and

branching ratios have been calculated with the help of the package FeynRules (version1.6.0) [159, 160], thus creating a new UFO model file in MadGraph5-aMC@NLO (version2.3.3) [168]. Using the full machinery of scalar mixing in this model, the decay widthsinto various channels have been obtained.

From Tables 5.3, 5.6 and 5.9, we see that decay (5.39) dominates, when the massesof H++

1 and H++2 are sufficiently separated. Also, when the phase space needed for this

decay (5.39) is not available, the process (5.36) dominates over all other remaining decays.

78


Benchmark points when decay (5.36) mostly dominates for H±±1 have been discussedin detail in the previous chapter. Here we supplement those observations with someresults for the case when decay (5.39) has an interesting consequence, as exemplitied byFigures 5.2 and 5.3.

Figure 5.2 specifically shows the effect of enhancement of the CP-violating phase. Wehave seen in Figure 5.1 that mH++

1− mH++

2goes up for α & 60. With this in view, the

plots in Figure 5.2 has been drawn for a case where both of the doubly-charged scalarshave appreciable coupling to same-sign dileptons. In the left panel of Figure 5.2, wherethe phase is lower, one notices two such dilepton pair peaks. The leptons selected forthis purpose satisfy:

∣∣∣pleptonT

∣∣∣ > 20 GeV, |η lep| < 2.5, |∆R ``| > 0.2 and |∆R `j| > 0.4 where∆R2 = ∆η2 +∆φ2. Each peak is the result of Drell-Yan pair production of the correspond-ing doubly-charged scalar and the presence of two triplets is clearly discernible from thepeaks themselves. In the right panel of Figure 5.2, however, with α = 65, one noticesonly the lower mass peak. This is because the decay H++

1 → H++2 h then reigns supreme.

As a result, one notices only lower mass peak, but events in association with an SM-likeHiggs are noticeable. Thus the signature of two triplets shows an interesting dichotomyof LHC signsls, depending on the value of the CP-violating phase.

Figure 5.3 further elaborates the first of the above situations. The plots there bear testi-mony to the situation where two peaks are still visible. Table 5.10 contains the numericalvalues of the number of events around each peak, for five of our benchmark spectra, withvarying phase. The numbers are illustrated for an integrated luminosity of 2500fb−1.The number of events corresponding to a bin within ±20GeV of the invariant mass peak.From this, we clearly expect several hundred events around the lower mass peak and60− 140 events around the higher one.

79


Figure 5.1: Variation of mass difference between (left panel) H++1 and H++

2 and (rightpanel) H++

1 and H+2 with phase of triplet α for BP 1 of scenario 1 and BP 3 of scenario 2

for α = 30, BP 4 of scenario 2 for α = 60

α = 30 Mass (GeV) BP 1 BP 2 BP 3 BP 4

H++1 516.61 513.83 522.13 537.62

Scenario 1 H++2 391.40 389.82 426.93 440.96

H+1 516.58 513.81 522.10 537.55

H+2 390.60 389.79 408.83 416.30

H++1 526.00 529.85 477.38 485.76

Scenario 2 H++2 397.61 390.10 389.10 389.33

H+1 525.94 529.80 477.34 485.70

H+2 393.79 390.01 389.00 389.28

H++1 558.71 562.35 485.76 477.38

Scenario 3 H++2 427.00 407.20 389.33 389.10

H+1 557.59 559.11 485.70 477.34

H+2 392.90 405.91 389.28 389.00

Table 5.1: Charged scalar masses for phase α = 30.

80



H01 730.57 726.65 738.36 760.19

Scenario 1 H02 730.49 726.62 738.32 760.15

H03 552.30 551.25 551.40 552.65

H04 552.15 551.20 551.34 552.56

h 125.16 125.18 125.20 125.15

H01 743.85 749.33 675.11 686.96

Scenario 2 H02 743.00 749.25 675.00 686.90

H03 552.21 551.50 550.17 550.53

H04 552.10 551.39 550.05 550.40

h 125.21 125.18 125.22 125.23

H01 787.10 789.25 687.00 676.15

Scenario 3 H02 787.00 789.10 686.90 676.08

H03 541.16 542.00 552.21 549.90

H04 541.07 541.91 552.00 549.75

h 125.23 125.26 125.13 125.16

Table 5.2: Neutral scalar masses for phase α = 30.

81


α = 30 Data BP 1 BP 2 BP 3 BP 4

BR(H++1 → H++

2 h) 5.1× 10−3 not allowed not allowed not allowedBR(H++

1 → H+2 W

+) 0.99 0.99 0.79 0.99

BR(H++1 → W+W+) 2.8× 10−3 6.5× 10−2 0.21 < 10−6

BR(H++1 → `+

i `+j ) < 10−20 < 10−19 < 10−17 < 10−22

Scenario 1 BR(H++2 → W+W+) 0.99 0.99 0.99 0.99

BR(H++2 → `+

i `+j ) < 10−17 < 10−17 < 10−16 < 10−19

σ(pp→ H++1 H−−1 ) 1.10 fb 1.13 fb 1.05 fb 0.42 fb

σ(pp→ H++2 H−−2 ) 3.97 fb 4.10 fb 2.70 fb 1.06 fb

BR(H++1 → H++

2 h) 0.84 0.96 not allowed not allowedBR(H++

1 → H+2 W

+) 0.13 0.03 0.76 0.42

BR(H++1 → W+W+) < 10−19 < 10−19 < 10−18 < 10−20

BR(H++1 → `+

i `+j ) 0.03 8.9× 10−3 0.24 0.58

Scenario 2 BR(H++2 → W+W+) < 10−18 < 10−18 < 10−19 < 10−18

BR(H++2 → `+

i `+j ) 0.99 0.99 0.99 0.99

σ(pp→ H++1 H−−1 ) 1.01 fb 1.02 fb 1.56 fb 1.43 fb

σ(pp→ H++2 H−−2 ) 3.60 fb 4.04 fb 3.97 fb 3.95 fb

BR(H++1 → H++


1 → H+2 W

+) 2.1× 10−3 1.3× 10−2 0.99 0.99

BR(H++1 → W+W+) < 10−13 < 10−13 < 10−9 < 10−10

BR(H++1 → `+

i `+j ) < 10−10 < 10−10 < 10−6 < 10−7

Scenario 3 BR(H++2 → W+W+) 0.03 0.01 0.04 0.02

BR(H++2 → `+

i `+j ) 0.97 0.99 0.96 0.98

σ(pp→ H++1 H−−1 ) 0.77 fb 0.74 fb 1.45 fb 1.58 fb

σ(pp→ H++2 H−−2 ) 3.61 fb 2.75 fb 3.95 fb 3.98 fb

Table 5.3: Decay branching ratios and production cross sections for doubly-chargedscalars for phase α = 30.

82



H++1 542.27 539.35 549.85 566.55

Scenario 1 H++2 406.60 405.20 438.46 450.96

H+1 542.22 539.20 548.73 564.92

H+2 405.90 405.07 422.28 428.94

H++1 543.30 538.15 551.62 564.34

Scenario 2 H++2 405.10 404.10 440.10 448.82

H+1 542.50 537.65 550.90 563.65

H+2 405.00 403.20 439.72 447.90

H++1 545.82 540.32 550.90 567.80

Scenario 3 H++2 409.80 405.00 439.50 452.45

H+1 544.71 539.46 550.15 565.90

H+2 409.00 404.75 425.38 432.80



H01 766.78 762.75 774.79 797.22

Scenario 1 H02 766.60 762.11 774.23 797.00

H03 573.00 572.50 575.50 576.17

H04 572.65 572.00 575.15 575.95

h 125.15 125.22 125.19 125.13

H01 768.10 760.37 772.90 795.85

Scenario 2 H02 768.00 760.13 772.75 795.50

H03 575.32 570.00 574.30 576.85

H04 575.15 569.22 573.78 576.20

h 125.12 125.16 125.24 125.17

H01 771.10 758.52 778.10 798.37

Scenario 3 H02 770.85 758.00 777.85 798.00

H03 577.31 568.75 578.29 577.21

H04 577.00 568.13 578.00 577.00

h 125.18 125.21 125.13 125.16


83



BR(H++1 → H++


1 → H+2 W

+) 8.2× 10−4 9.1× 10−4 0.90 0.96

BR(H++1 → W+W+) 2.4× 10−5 1.7× 10−4 0.09 0.04

BR(H++1 → `+

i `+j ) < 10−21 < 10−21 < 10−18 < 10−19

Scenario 1 BR(H++2 → W+W+) 0.99 0.99 0.99 0.99

BR(H++2 → `+

i `+j ) < 10−18 < 10−18 10−18 < 10−17

σ(pp→ H++1 H−−1 ) 0.88 fb 0.87 fb 0.80 fb 0.71 fb

σ(pp→ H++2 H−−2 ) 3.39 fb 3.33 fb 2.43 fb 2.13 fb

BR(H++1 → H++


1 → H+2 W

+) 4.5× 10−4 3.9× 10−4 0.64 0.88

BR(H++1 → W+W+) < 10−20 < 10−21 < 10−19 < 10−20

BR(H++1 → `+

i `+j ) 3.1× 10−6 1.7× 10−4 0.36 0.12

Scenario 2 BR(H++2 → W+W+) < 10−19 < 10−18 < 10−18 < 10−19

BR(H++2 → `+

i `+j ) 0.99 0.99 0.99 0.99

σ(pp→ H++1 H−−1 ) 0.93 fb 0.89 fb 0.86 fb 0.75 fb

σ(pp→ H++2 H−−2 ) 2.55 fb 3.35 fb 2.46 fb 2.18 fb

BR(H++1 → H++


1 → H+2 W

+) 3.6× 10−5 1.4× 10−4 0.99 0.99

BR(H++1 → W+W+) < 10−13 < 10−12 < 10−8 < 10−9

BR(H++1 → `+

i `+j ) < 10−10 < 10−10 < 10−10 < 10−8

Scenario 3 BR(H++2 → W+W+) 0.02 0.04 0.97 0.05

BR(H++2 → `+

i `+j ) 0.98 0.96 0.03 0.95

σ(pp→ H++1 H−−1 ) 0.92 fb 0.95 fb 0.84 fb 0.73 fb

σ(pp→ H++2 H−−2 ) 3.42 fb 3.37 fb 2.44 fb 2.15 fb


84



H++1 557.90 563.51 564.20 556.56

Scenario 1 H++2 412.20 411.51 434.37 439.71

H+1 557.62 563.25 559.18 548.00

H+2 411.65 411.18 423.27 426.15

H++1 558.20 565.20 566.40 554.30

Scenario 2 H++2 411.90 413.61 436.56 438.12

H+1 558.00 564.50 565.90 553.65

H+2 410.75 412.85 435.85 435.32

H++1 556.65 560.30 567.80 552.90

Scenario 3 H++2 410.25 408.35 437.90 436.59

H+1 556.00 559.75 563.21 550.00

H+2 409.85 407.80 425.56 429.11



H01 788.52 796.91 784.64 765.05

Scenario 1 H02 788.35 796.27 784.21 764.62

H03 581.43 583.16 579.62 577.78

H04 581.32 582.95 579.13 577.21

h 125.16 125.24 125.14 125.20

H01 790.21 793.82 786.52 762.90

Scenario 2 H02 790.00 793.11 786.09 762.42

H03 579.32 580.16 582.32 574.21

H04 579.00 579.92 582.00 573.86

h 125.15 125.10 125.21 125.09

H01 786.51 790.63 785.00 760.71

Scenario 3 H02 786.00 790.27 784.32 760.29

H03 577.82 576.21 580.14 570.90

H04 577.50 576.00 579.55 570.58

h 125.23 125.14 125.23 125.18


85



BR(H++1 → H++

2 h) 0.99 0.98 0.99 not allowedBR(H++

1 → H+2 W

+) 3.9× 10−4 2.6× 10−2 0.01 0.94

BR(H++1 → W+W+) 1.7× 10−5 8.9× 10−3 6.9× 10−5 0.05

BR(H++1 → `+

i `+j ) < 10−21 < 10−20 < 10−21 < 10−19

Scenario 1 BR(H++2 → W+W+) 0.99 0.99 0.99 0.99

BR(H++2 → `+

i `+j ) < 10−18 < 10−18 < 10−19 < 10−14

σ(pp→ H++1 H−−1 ) 0.79 fb 0.71 fb 0.72 fb 0.84 fb

σ(pp→ H++2 H−−2 ) 3.22 fb 3.15 fb 2.48 fb 2.51 fb

BR(H++1 → H++

2 h) 0.99 0.79 0.99 not allowedBR(H++

1 → H+2 W

+) 3.2× 10−5 0.21 3.1× 10−3 0.88

BR(H++1 → W+W+) < 10−21 < 10−21 < 10−21 < 10−21

BR(H++1 → `+

i `+j ) 2.1× 10−4 1.4× 10−4 3.7× 10−4 0.12

Scenario 2 BR(H++2 → W+W+) < 10−18 < 10−19 < 10−17 < 10−18

BR(H++2 → `+

i `+j ) 0.99 0.99 0.99 0.99

σ(pp→ H++1 H−−1 ) 0.77 fb 0.74 fb 0.81 fb 0.86 fb

σ(pp→ H++2 H−−2 ) 3.26 fb 3.19 fb 2.46 fb 2.53 fb

BR(H++1 → H++

2 h) 0.99 0.90 0.99 not allowedBR(H++

1 → H+2 W

+) 2.5× 10−4 0.10 1.2× 10−2 0.99

BR(H++1 → W+W+) < 10−14 < 10−13 < 10−10 < 10−10

BR(H++1 → `+

i `+j ) < 10−10 < 10−11 < 10−12 < 10−8

Scenario 3 BR(H++2 → W+W+) 0.03 0.04 0.89 0.02

BR(H++2 → `+

i `+j ) 0.97 0.96 0.11 0.98

σ(pp→ H++1 H−−1 ) 0.81 fb 0.75 fb 0.72 fb 0.83 fb

σ(pp→ H++2 H−−2 ) 3.28 fb 3.20 fb 2.50 fb 2.54 fb


86


Figure 5.2: Invariant mass distribution of same sign di-leptons for (left panel) α = 60 and(right panel) α = 65 for BP 4 of Scenario 2

Benchmark points, Scenario 2 No. of events at the H++2 peak No. of events at the H++

1 peak

BP 3 for α = 30 527 100

BP 4 for α = 30 520 139

BP 3 for α = 45 329 65

BP 4 for α = 45 287 60

BP 4 for α = 60 389 72

Table 5.10: Number of same-sign dilepton events generated at the LHC, for the bench-mark points corresponding to Figure 3. The integrated luminosity is taken to be 2500fb−1,for√s = 13TeV .

87


Figure 5.3: Invariant mass distribution of same sign di-leptons for chosen benchmarkpoints. In the top (left panel) BP 3 and (right panel) BP 4 of Scenario 2 for α = 30. In themiddle (left panel) BP 3 and (right panel) BP 4 of Scenario 2 for α = 45. In the bottom,BP 4 of Scenario 2 for α = 60.

88

5.5. CONCLUSIONS

5.5 Conclusions

We have considered a one-doublet, two-triplet Higgs scenario, with one CP-violatingphase in the potential. It is noticed that a larger phase leads to bigger mass-separationsbetween the two doubly-charged mass eigenstates, and also between the states H++

1 andH+

2 . Consequently, this scenario admits a larger region of the parameter space, whenthe decay H++

1 → H++2 h opens up. When it is allowed, this decay often overrides

H++1 → H+

2 W+. While the role of the latter decay as a characteristic signal of such mod-

els was discussed in our earlier work, we emphasize here that the former mode leads toanother interesting signal, arising from H++

1 → `+i `

+j h. This would mean that the pro-

duction of SM-like Higgs together with same-sign dileptons peaking at the mass of thelighter doubly-charged scalar. Such a signal, too, may give us a distinctive signature of atwo-triplet scenario at the LHC.

89


90

Chapter 6

Dark matter

6.1 Introduction

Astrophysical observations during the past few decades produced evidences beyonddoubt that most of the matter consisting our universe is non-luminous or dark. The ex-istence of this dark matter (DM) is one of the strongest indications that there should bephysics beyond the Standard Model (SM) of particle physics. Numerous indirect obser-vations in the astronomical and cosmological scales [169] such as, accurate measurementof galactic rotation curves, measurements of orbital velocities of individual galaxies inclusters, cluster mass determination via gravitational lensing, precise measurements ofthe Cosmic Microwave Background (CMB) acoustic fluctuations, the abundance of lightelements and the mapping of large-scale structures point to the presence of new elemen-tary particles that interact only through the gravitational and (perhaps) weak forces. Inaddition cosmological simulations based on ΛCDM model [170] successfully predict theobserved large-scale structures of the Universe. The Planck satellite mission [171] haspublished precise measurements of the CMB, which are in complete agreement with thepredictions of ΛCDM model, describing a cosmos dominated by dark energy (Λ) and cold(i.e non-relativistic) dark matter (CDM). According to the ΛCDM model, which providesthe only platform that can explain all observations till date, our universe is composed of5% atoms (known matter), 27% dark matter and 68% dark energy.

The first evidence for a dark matter dominance of the Coma galaxy cluster was indi-cated by astronomer Fritz Zwicky in 1933 [172]. Despite the advancement of our under-standing of the amount and distribution of dark matter on various scales, we still don’thave any answer to the most basic question – What is the dark matter made of? One pos-sible answer is that, it is made of new particles and one must look beyond the standard

91

CHAPTER 6. DARK MATTER

paradigm of particle physics to search for them. As a consequence, several more ques-tions arise, like — Does the dark matter consists of one particle species or involve many? Whatare the properties of these particles, such as their maases, interaction cross-sections, spin and otherquantum numbers? Are these particles absolutely stable or very long-lived (i.e. have half-lifelonger than the age of the Universe)? [173] All these are open questions waiting to be ex-plored. In the next section we briefly point out some evidences for the existence of darkmatter (DM).

6.2 Evidences for dark matter

In this section we will briefly discuss some of the compelling evidences that consolidatethe presence of DM in the universe.

• Galactic rotation curves : It has been mentioned earlier that the evidence for DMwas first discovered by Zwicky. Measurements of the velocity dispersion of galaxiesin the Coma cluster led to the conclusion that they could not be bound to the clusterby the gravitational attraction of visible matter (stars, gas, dust) alone.

Existence of DM was again pointed out in the 1970’s by Rubin and others [174]. Thisis, perhaps, the most convincing evidence for DM till date. The evidence came fromthe observation that various luminous objects (stars, gas clouds, globular clusters orentire galaxies) move faster than the expectation, if they only felt the gravitationalattraction of other visible objects. An important step in this regard was the measure-ment of galactic rotation curves. The rotational velocity v of an object, as a functionof the radius r beyond the visible core of galaxy, indicates that the mass continues togrow with the radius. If all the mass of the galaxy M is assumed to be concentratedat the core, an object of massm orbiting the galaxy would experience a gravitationalforce F = GMm/r2 = mv2/r, and v = (GM/r)1/2 i.e v(r) ∝ 1/

√r. Instead, in most

galaxies it has been found that v becomes approximately constant up to the largestvalues of r where the rotation curve can be measured. In our galaxy, v ' 240km/s

at the location of our solar system, with little change up to the largest observableradius. This observation can be explained by assuming the existence of a dark halo,with mass density ρ(r) ∝ 1/r2 i.e M(r) ∝ r. This indicates that galaxies containmore mass than the total amount of masses in stars, gas and other visible objects.

• Gravitational lensing and Bullet cluster : Gravitational lensing is the bending (orlensing) of light due to the presence of massive objects. This is a general relativis-

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6.2. EVIDENCES FOR DARK MATTER

tic phenomenon. Galaxy clusters being highly massive structures show this effect.Due to this effect, a background object seems brighter than it otherwise would ap-pear (see, e.g., [175, 176]). The dark matter present in the galaxy cluster, though notvisible, contributes significantly to the total mass of the cluster. The best evidence todate for the existence of DM comes from the weak lensing observations of famousBullet Cluster (1E0657-558), a unique cluster merger [177], where the lensing effectshows a large amount of dark matter. In addition to this, another property of DMcomes out of this cluster merger, the DM halos have passed right through both thegas clouds and appear almost undisturbed after the collision. But the visible gasclouds of both clusters have undergone characteristic changes. It becomes evidentfrom this, that the DM interacts with luminous matter as well as itself very weakly.Also the observation that the lensing effect extends beyond the central (visible) re-gion of the cluster may be interpretated as a pointer to the non-baryonic nature ofdark matter.

• Other observations : There are many other observations which establishes the ex-istence of DM on a stronger footing. One of the most important phenomenonis the determination of cosmological parameters by the power spectrum of CMBaniosotropy. The best-fit values to the ΛCDM model gives matter density of theuniverse as ΩMh

2 = 0.127+0.007−0.013 and primordial baryon density of the universe as

ΩBh2 = 0.0223+0.0007

−0.0009 [178]. The difference in these two densities clearly indicatesthat most of the matter component in the universe is not the atoms but somethingelse, thus confirming the existence of DM. This also mandates that the DM is neces-sarily non-baryonic. Another important way to determine the baryon density of theuniverse is based on Big-Bang Nucleosynthesis (BBN). The baryon density is consis-tent with the data obtained from CMB power spectrum, ΩBh

2 = 0.0216+0.0020−0.0021 [179].

A new technique to determine matter density ΩM uses large-scale structure via thepower spectrum in galaxy-galaxy correlation function. As a result of the acousticoscillation in the baryon-photon fluid, the power spectrum also shows the "baryonoscillation" [180]. Independent of CMB results, they determined the matter densityto be ΩMh

2 = 0.130± 0.010. This is consistent with the CMB data and thus stronglyadvocates the need for non-baryonic dark matter.

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6.3 Some properties of dark matter

The observations that give us evidence for DM also shed light on some of the obviousproperties of DM. A theory hoping to solve the DM riddle is also guided by these require-ments. We enlist below some of the properties of DM indicated by the experminents.

• The DM must be ’dark’, i.e., it should have no (or extremely weak) interactions withphotons. So it must be electrically neutral. If DM had either a small charge or a smallelectric or magnetic dipole moment, it would couple to the photon-baryon fluid andthus altering the features of the CMB as well as the matter power spectrum.

• The DM cannot be baryonic. The interaction between DM and baryons should alsobe weak. A consequence of non-baryonicity is color neutrality, meaning DM parti-cles may not take part in strong interactions, otherwise that particle would just actas baryonic matter. As mentioned above, observations of baryonic acoustic oscilla-tion (BAO) and CMB angular power spectrum disfavour the non-baryonic natureof DM.

• Weak interactions of DM particles are not excluded. But, coupling to the elec-troweak gauge bosons W± and Z must be somewhat weaker than that of other SMparticles. Otherwise, direct-detection experiments would already have found suchinteractions.

• The stability of at least the lightest DM particles must be high and this in turnpredicts the DM self-interactions to be rather weak. The lack of impact of cosmicevents that influence the baryonic matter distribution, such as the merger of theBullet cluster, on the DM-distribution imply that the DM particles have to be highlycollision-less. Even if the particles in the dark-sector have no coupling to the knownSM particles, interactions between particles in the dark-sector eventually affect thestructures of DM-halos. For hard-sphere elastic scattering, the constraints are atthe level of cross-section per unit dark matter mass σ/mχ . 1cm2/g obtained fromobservations of the structure of galaxy clusters [181].

• As already mentioned, the DM must be stable. To put it another way, DM musthave lifetime larger than the age of the universe.

• Studies on the anisotropies of the CMB radiation spectrum requires the DM to be"cold" that is to say, something that was non-relativistic at the time of decoupling.

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6.4. CANDIDATES FOR DARK MATTER

keV-range warm dark matter, too is not ruled out.

• DM can not be made up of SM particles as most of them are charged. Within theambit of SM only neutrinos can be a potential DM candidate. Actually it does con-tribute in the relic density of DM to a small extent. But it can not be the CDMcomponent, as neutrinos are expected to have masses in sub-eV range [182]. Withthis small mass it can not probably contribute significantly to the matter densityof universe. From the Gunn-Tremaine bound [183] it appears that massive galactichalos can not be made up of neutrinos of mass ≤ 1 MeV.

Thus it becomes evident that the SM can not accommodate a viable DM candidate. But itis also worth noting that although observations tend to imply that DM is non-baryonic,there are exotic baryonic objects e.g., white dwarfs, neutron stars, super-massive blackholes which comprise MAssive Compact Halo Objects (MACHO) [184, 185], which mayin principle still contribute partially to the DM content.

6.4 Candidates for dark matter

We outline below some potential candidates for DM :

• WIMPs : Weakly Interacting Massive Particles(WIMPs), denoted as χ, are particleswith mass roughly between 10 GeV and few TeV and cross-sections of approxi-mately weak interaction strength. Within standard cosmology, their present relicdensity can be calculated reliably if the WIMPs were in thermal and chemical equi-librium with the SM particles after inflation. From particle physics perspective, theearly universe was a high energy place where energy and mass could switch fromone form to the other according to Einstein’s equation E = mc2. Pairs of particlesand their antiparticles are being created and annihilated. However, as the universeexpands, it cools and as a result, it loses the energy necessary to create particle-antiparticle pairs. For a particular particle, the creation process depends on themass of the particle — the more is the mass, the more energy is required to producethem. As the temperature decreases the massive particle-antiparticle pairs "freezeout" earlier. After freezing out the remaining particle-antiparticle pairs can mutu-ally annihilate leaving only energy. To avoid this fate, there must either be someasymmetry or the "cross section" - the probability for interacting - must be so lowthat particles and their antiparticles move in different directions without meeting

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often enough to annihilate completely. This process leaves some relic density thatdepends on the properties of the particles.

To match the relic density necessary to make up the cosmic dark matter, the crosssection required is about that of the weak nuclear force. A particle that interactsthrough the weak force but not the electromagnetic force will have the right relicdensity. Moreover, it won’t interfere with BBN or the CMB. The hypothesizedWIMPs fit the requirements for cosmologists’ ’Cold Dark Matter’. This coincidenceof scales - the relic density and the weak force interaction scale - is sometimes re-ferred to as the "WIMP miracle" and was part of the motivation to adopt the WIMPas the leading candidate for cosmological dark matter.

• Axion : Although its not WIMP type DM, but axion is one of the leading non-baryonic cold DM candidates. The idea of axion, a light pseudoscalar boson, wasput forward to solve the strong CP problem [186]. A number of astrophysical obser-vations and laboratory experiments put bound on axion mass to be∼ 10−9 eV. Eventhough the mass is so small they are still cold as their production is non-thermal.Some reviews and recent studies can be found in [187–193].

• The lightest supersymmetric particle : By far the most popular and extensivelystudied BSM scenario is Supersymmetry (SUSY). In supersymmetric theories everySM particle has its super-partner particle differing in spin by half a unit. In most ver-sions of SUSY, there is a conserved discrete symmetry, R-parity, to fulfill the require-ment of proton stability.It is defined as, R = (−1)3(B−L)+2S . Here B and L is baryonand lepton number respectively and S is the spin of the particle. For SM particles R= 1 and for supersymmetric particles R = -1. In the minimal supersymmetric stan-dard model (MSSM), four linearly independent, non-strongly interacting neutralspin 1/2 particles occur, and are called neutralinos. These are linear superpositionsof the superpartners of the photon, the Z-boson and the two neutral scalars. Thelightest neutralino is often the lightest SUSY particle (LSP) and is a good DM candi-date. A number of references on neutralino CDM can be found in literature. Someof the studies on neutralino LSP can be found in references [194–197]. However,there are other variants of SUSY model where the DM candidate is not neutralinobut some other supersymmetric particle (sneutrino or gravitino for example).

• Inert Higgs Doublet Model : Two Higgs doublet model (2HDM) is a simple exten-sion of SM. In 2HDM there is one extra Higgs doublet in addition to the SM Higgs

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6.5. CALCULATION OF DARK MATTER RELIC DENSITY

doublet. This model is attractive because it is a minimal extension to SM whichcan accommodate additional source of CP violation which is needed to explain thebaryon asymmetry of the universe. Also there are other motivations. In a versionof 2HDM, named as inert (Higgs) doublet model (IDM) [198] an unbroken Z2 sym-metry ensures the stability of the lightest neutral scalar (or pseudoscalar) particle.A recent study of this DM candidate can be found in [199].

• Universal Extra Dimension : Another theory that produces a viable dark mattercandidate is Universal Extra Dimensions (UED), in which the existence of addi-tional compact spacelike dimension is postulated. The extra dimension is not visi-ble because its compactification radius is too small to be observed. The dark mattercandidate in UED theories is a stable particle known as the Lightest Kaluza- KleinParticle or LKP which is odd under a Z2 symmetry called Kaluza-Klein parity. Var-ious aspects of LKP as a dark matter has been discussed in [200–211].

• Heavy neutrinos : Another possible candidate for DM is heavy neutrino. SM neutri-nos cannot constitute a significant part of DM because current upper limits on theirmasses imply that they would freely stream on scales of many megapersecs (Mpc)and hence wash out the density fluctuations observed at these scales. Cosmologicalsimulations have shown, however, that a Universe dominated by neutrinos wouldnot be in agreement with the observed clustering scale of galaxies [212].

Sterile neutrinos are hypothetical particles which were originally introduced to ex-plain the smallness of the neutrino masses [213]. In addition, they provide a viabledark matter candidate. Depending on their production mechanism, they wouldconstitute cold or a warm dark matter candidate [214, 215]. However, the neutrinomust be stable and it is not understood that why a massive neutrino should not beallowed to decay.

6.5 Calculation of dark matter relic density

After its production WIMPs (χ) remain in thermal equilibrium and in abundance whenthe temperature of the universe is greater than the mass of the particle mχ. The equilib-rium abundance (or density) of these particles is maintained by the annihilation of theseparticles with their anti-particles, χ into other lighter particles (say X), i.e χχ −→ XX

and vice versa XX −→ χχ. But the temperature of the universe decreases gradually andwhen it becomes less than mχ, the equilibrium abundance drops. Clearly the annihilation

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rate for χχ −→ XX will also drop. At the point when the annihilation rate falls below theexpansion rate of the universe H, the interactions maintaining the thermal equilibrium’freeze out’. Since the χ ’ s are stable (i.e., they can not decay) then after the freeze outthe abundance of them also become fixed and what remains afterward is just the thermalrelic of WIMPs. In the next subsection we will describe the calculation of the relic densityfollowing standard equilibrium thermodynamics [216].

6.5.1 Standard calculation of relic density

The number density of the particle χ, in thermal equilibrium of the early universe is givenby,

neq =g

(2π)3

∫d3pf(~p), (6.1)

where g is the internal degrees of freedom of χ and f(~p) is the phase space distributionfunction and is given by usual Fermi-Dirac (FD) or Bose-Einstein (BE) distribution,

f(~p) =1

exp(E−E0

T

)± 1

, (6.2)

where ’+’ is for FD statistics and ’-’ for BE statistics andE0 is a constant of the correspond-ing species.In the relativistic limit (T m) and T E0,

neq =

(ζ(3)π2

)gT 3 for BE statistics,(

3ζ(3)4π2

)gT 3 for FD statistics,

(6.3)

here ζ is the Riemanian zeta function. In the non-relativistic limit (m T ), both for FDand BE species, and we have,

neq = g

(mT

2π

)3/2

exp

(m− E0

T

). (6.4)

The evolution with time (or temperature), of the number density of χ is given by theBoltzmann equation,

dn

dt= −3Hn− 〈σv〉(n2 − n2

eq), (6.5)

whereH =

1

a

da

dt(6.6)

is the Hubble parameter (a is the scale factor of the universe) and 〈σv〉 is the thermally av-eraged annihilation cross section times relative velocity. The first term in the right-hand

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6.5. CALCULATION OF DARK MATTER RELIC DENSITY

side of equation. 6.5 accounts for the reduction of number density due to the expansion ofthe universe. The second term takes care of decrease (or increase) in n due to the interac-tion of χ with other particles in the spectrum. After solving equation 6.5 we can calculatethe contribution of χ in the energy density of the universe by defining the quantity, Ωχ as,

Ωχh2 ≡ ρχ

ρc=mχn

ρc, (6.7)

where h is the Hubble parameter in the units of 100 km s−1 Mpc−1 and ρc is the criticaldensity of the universe and is given by,

ρc =3H2

8πGN

. (6.8)

At the point when the annihilation rate Γ = n〈σv〉 . H , the annihilation of χs ceases andthe relic abundance remains fixed afterward. At the freeze out temperature the annihila-tion cross section can be expanded in powers of squared relative velocity,

σv = a+ bv2 + . . . , (6.9)

where the first term comes from the s-wave annihilation and the second from both s andp-wave annihilation. A nice discussion on these types of annihilation cross section inthe low velocity limit can be found in [217]. In most of the cases the first two terms inthe expansion are enough to produce a fair estimate of the relic density. With properapproximations equation. 6.5 can be solved analytically and the relic density is givenby [216, 218],

Ωχh2 ≈ 1.04× 109/1GeV

MPl

xF√g∗(xF )

1

a+ 3bxF

, (6.10)

where MPl is the Planck mass and g∗(xF ) is the total number of relativistic DOF at thefreeze out temperature. Here xF (= m/TF , TF being the freeze out temperature) is solvedfrom the equation,

xF = ln

(15

8

√5

2

g

2π3

mMPl(a+ 6b/xF )√g∗(xF )xF

). (6.11)

However, there are three important exceptions to the validity of the above mentionedprescription [219, 220],

• annihilation near mass thresholds (i.e., kinematically forbidden channels at T = 0,but which can be significant at higher temperatures),

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• coannihilations (i.e., when particles which are slightly heavier than χ and affect thenumber density of χ), and

• resonances in the annihilation cross section (i.e., when mχ is half the mass of theparticle exchanged in the s-channel annihilation process).

Systematic treatment of finite temperature corrections takes care of the first case. Butthis marginally affects the relic density. The latter two cases depend significantly on theparticle spectrum and thus on the parameters of the theory.

6.5.2 Coannihilation

In the particle spectrum of the theory if there are particles nearly degenerate with the relicparticle then the freeze out of these particles occurs almost at the same epoch when therelic particle χ decouples and can affect the relic abundance of χ. For example, considerthere are N particle species, χi (i = 1, 2, . . . , N) and mi < mj if i < j, i.e., χ1 is the light-est particle. The number density ni of each species χi will obey appropriate Boltzmannequations. Moreover, all the heavier particles (χi with i > 1) will ultimately decay to χ1.So from any number Ni of χi particles we end up getting exactly Ni number of χ1 at theend of the decay chain. Thus to determine the relic abundance of χ1 it is meaningful tostudy the evolution of the number density n(=

∑j nj) instead of each number density

separately. In this case the Boltzmann equation reduces to,

dn

dt= −3Hn− 〈σeffv〉(n2 − n2

eq). (6.12)

The quantity σeff is given by,

σeff(x) =1

g2eff

N∑i,j=1

σijFiFj (6.13)

where

geff(x) =N∑i=1

Fi(x), (6.14a)

Fi(x) = gi(1 + ∆i)3/2 exp(−x∆i), (6.14b)

with

∆i =mi −m1

m1

and x =m

T. (6.15)

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6.6. SEARCH FOR DARK MATTER

Here σij ≡ σ(χiχj → SM) and gi is the number of internal degrees of freedom of thespecies χi taking part in the annihilation or coannihilation process. In the non-relativisticlimit we have, 〈σeffv〉 ∼ aeff(x) + beff(x)v2 + O(v4). The approximate expression for therelic density will now become,

Ωχh2 ≈ 1.04× 109/1GeV

MPl

xF√g∗(xF )

1

Ia + 3IbxF

, (6.16)

where Ia,b are given by,

Ia = xF

∫ ∞xF

aeff(x)x−2dx, (6.17a)

Ib = 2x2F

∫ ∞xF

beff(x)x−3dx. (6.17b)

The freeze out temperature is given by,

xF = ln

(15

8

√5

2

geff(xF )

2π3

m1MPl(aeff(xF ) + 6beff(xF )/xF )√g∗(xF )xF

). (6.18)

We shall now briefly discuss some efforts for the search of DM in the next section.

6.6 Search for dark matter

Search for the DM particles has become one of the most exciting topics in Astroparticlephysics and considerable progress has been achieved in experimental and theoretical re-search. Various efforts have been made and are still going on to explore the mystery ofDM and presently they can be divided into three categories : indirect searches, acceler-ator searches and direct searches. Indirect methods employ space-based and terrestrialinstruments to search for signals from products of WIMP annihilation, such as γ-rays orpositrons. Accelerator searches such as those at the LHC look for WIMPs leading to miss-ing reconstructed energy in the final states to get a hint of the DM mass. However, neitherof these techniques is by itself capable of providing a robust statement on the nature ofDark Matter. Instead, it is direct observation of WIMPs in terrestrial experiments that canbe a definitive way of detection. Meeting this challenge and detecting galactic WIMPs isrecognised as one of the highest priorities in physics today.

6.6.1 Direct detection

Signature of dark matter in a direct detection experiment consists of recoil spectrum ofsingle scattering events. Here, the aim is to identify nuclear recoil produced by the col-

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lisions between the new particles and the detector’s target nuclei. The elastic scatteringof WIMPs with masses of (10 − 1000)GeV/c2 would produce nuclear recoils in the rangeof (1− 100)keV [221]. To identify such low-energy interactions clearly, a detailed knowl-edge on the signal signatures, the particle physics aspects and nuclear physics modellingis mandatory. Furthermore, for the calculation of event rates in direct detection experi-ments, the dark matter density and the halo velocity distribution in the Milky Way arerequired. In the next paragraph we mention results of some direct detection experiments.For a deeper understanding on direct detection experiments, see [222].

The CoGeNT (Coherent Germanium Neutrino Technology) detector [223–225] begancollecting data at the Soudan Underground Laboratory, Minnesota, in 2009. They haverevealed a seasonal modulation consistent with the presence of WIMPs with masses' 7GeV/c2. The CDMS (Cryogenic Dark Matter Search) [226] and its successor theCDMS II experiment [227] use germanium and silicon bolometers to search for darkmatter. The experiments are located at the Soudan Underground Laboratory. A com-bined analysis of all CDMS II detectors yields an upper limit on the WIMP-nucleon spin-independent cross-section of 3.8× 10−44cm2 for a WIMP mass of 70GeV/c2. CDMS II per-formed additionally a study for an annual modulation of the event rate using data fromOctober 2006 to September 2008 [228]. No evidence for an annual modulation was foundand this data disfavours the modulation claim of the CoGeNT experiment. The succes-sor of the CDMS II experiment is the SuperCDMS detector which employs an improvedtechnology. SuperCDMS experiment focuses on dark matter masses below 30GeV/c2 andthey have derived a limit on the cross-section for a 8GeV/c2 WIMP mass which is given by1.2× 10−42cm2 [229]. The results of SuperCDMS set most sensitive exclusion limits at lowWIMP masses [230]. SuperCDMS is also the first direct detection experiment which de-rives limits on more general WIMP interactions calculated with a non-relativistic effectivefield theory [231]. The second generation of the SuperCDMS experiment will be locatedat SNOLAB. A similar detector concept is used by the EDELWEISS collaboration [232]which operates at Laboratoire Souterrain de Modane (LSM). The most sensitive limit canbe derived at a WIMP mass of 90GeV/c2 and a cross-section of around 4 × 10−44cm2.Both the CDMS and the EDELWEISS experiments have performed axion searches also.In the CRESST-II experiment [233, 234] at Laboratori Nazionali del Gran Sasso (LNGS),an excess of events is observed, corresponding to a WIMP mass of 11.6GeV/c2(4.2σ) or25.3GeV/c2(4.7σ) with a cross-section of 3.7× 10−41cm2 or 1.6× 10−42cm2 respectively. Infuture, the CRESST collaboration is aimed to focus on low mass WIMP. This would allowto search for WIMP masses down to 1GeV/c2. A possible next generation experiment,

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EURECA (European Underground Rare Event Calorimeter Array) [235], is a joint effortmostly originating from the EDELWEISS, CRESST and ROSEBUD [236] collaborations.The final goal is to reach a sensitivity of 3 × 10−46cm2. In principle, a joint experimentbetween EURECA and SuperCDMS would be feasible, combining various technologiesand exploiting their complementarity. From the DarkSide experiment [237] an exclu-sion limit is placed which is at 6.1× 10−44cm2 at 100GeV/c2 WIMP mass. The XENON10experiment obtained sensitivities at WIMP masses as low as 5GeV/c2 [238]. The suc-cessor of XENON10, XENON100 [239, 240], started operation at the LNGS laboratory in2009. However, no evidence for dark matter was found. While interpreting the data asspin-independent interactions of WIMP particles, a best sensitivity of 2 × 10−45cm2 for55GeV/c2 WIMP mass was obtained. Furthermore, exploiting the low background rateof the experiment, various leptophilic dark matter models have been excluded [241]. Toincrease the sensitivity further, a next generation detector, XENON1T [242], is being con-structed. The goal is to achieve two orders of magnitude improvement in sensitivity byreducing the background by a factor of ∼ 100 compared to XENON100. In 2013, the LUX(Large Underground Xenon) experiment [243], installed at the Sanford underground lab-oratory in the US, released first data [244]. These results improved towards the results ofXENON100 down to 7.6 × 10−46cm2 for a WIMP mass of 33GeV/c2. This is currently thelowest limit for direct detection experiments for spin independent interactions for WIMPmasses above 6GeV/c2. The LUX and ZEPLIN collaborations have joined to build themultiton LZ detector [245] to increase the sensitivity on WIMP-matter cross-sections. An-other experiment, PandaX [246], tested cross-sections down to 10−44cm2 for a 45GeV/c2

WIMP mass [247]. One of the first bounds on the dark matter cross-section from a detec-tor using superheated fluids was achieved by the SIMPLE experiment which is operatedat LSBB in France. A sensitivity to the spin-dependent WIMP-proton cross-section of5.7 × 10−39cm2 at 35GeV/c2 was achieved [248]. PICO detector [249] (formed from thePICASSO and COUPP experiments) shows the strongest exclusion limit on the spin-dependent WIMP-proton cross-section at 40GeV/c2 of around 9 × 10−40cm2 at 90%C.L.PICO is operated in the SNOLAB underground laboratory.

6.6.2 Indirect search

While direct detection experiments discussed in the previous section require dark matterto be present in the Earth’s neighborhood, this is not generally necessary for indirectdetection searches. In this case, it is assumed that WIMPs are their own anti-particles and

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will annihilate in standard model particles after collisions occur between them. Possibleannihilation scenarios include

χχ −→ qq, `¯,W+W−, ZZ. (6.19)

These primary particles eventually decay into positrons, electrons, anti-protons, protons,neutrinos and γ-rays, which can be observed by suitable detectors. As it is generallynot known which particles are preferentially generated in the WIMP annihilation, var-ious channels are usually considered in the analysis independently. The Sun is a veryinteresting target for such searches, as it is expected to store WIMPs in its core whilemoving through the galactic halo. Results from the indirect detection of signals fromWIMP annihilation in the Sun can be directly compared to the results from undergroundexperiments [250].

An analysis from Super-Kamiokande places the first constraints from indirectsearches on spin-dependent scattering cross sections below dark matter mass ∼10GeV/c2 [251]. A Monte Carlo study shows that no events have been observed, exclud-ing spin-dependent WIMP-nucleon cross sections above 10−39 to 10−40cm2. Similar resultson WIMP annihilations in the Sun from other neutrino detectors have been observed fromIceCube [252] and ANTARES [253] experiments. Both experiments did not detected anysignal. In particular ANTARES, which is located in the northern hemisphere, mainly fo-cuses on dark matter annihilation signals from the Galactic Centre. IceCube, which isa gigantic neutrino detector installed in the South Pole, is also searching for WIMP an-nihilation in the Galactic Center. Preliminary studies show that IceCube is sensitive tovelocity-averaged annihilation cross sections of ≈ 10−21 to 10−22cm3s−1 for WIMP massesas low as 30GeV/c2 [254,255]. The ’smoking gun’ signature for WIMP annihilation wouldbe a line in the γ-spectrum of objects expected to have an high WIMP density, e.g., theGalactic Center or dwarf spheroidal galaxies. As WIMPs do not couple directly to pho-tons, the processes χχ −→ γγ or χχ −→ γZ do not happen at tree-level. They appearonly as largely suppressed second-order processes. Still, the two monoenergetic gammaswould produce a sharp, distinct spectral feature at the dark matter mass (mχ), accom-panied by a somewhat broader and lower peak from χχ −→ γZ at reduced energies.The FermiLAT (Fermi satellite - Large Area Telescope) observed the γ-spectrum from 20MeV to 300 GeV. In 2012, there has been a claim that a γ- line at 130GeV has been foundin the LAT data at the Galactic Center [256, 257]. In an updated analysis the global sig-nificance at 130GeV was found to be only 1.6σ and this seemd to be too wide for theinstrument’s resolution. At this point, it is thought that more data is needed to varify

104


the situation. PAMELA experiment reported a rising fraction of positrons in the totale−e+-flux for energies above ∼ 5 GeV [258, 259]. An excess of antiparticles, in particu-lar positrons, is widely discussed as a promising signature for dark matter annihilation.Moreover, the AMS-02 experiment recently confirmed the excess, which keeps increasingup to energies of ∼ 300 GeV [260, 261], the current high-energy limit of the instrument.However, several difficulties with the interpretation of the positron excess observed byPAMELA and AMS-02 have been pointed out and at present these results are consideredto be somewhat controvesial.

6.6.3 Accelerator search

WIMPs are very weakly interacting. So, they do not deposit any energy in the detectorsand one needs to look at the total energy and momentum of an event in order to inden-tify such particles via a missing energy signal. In pp-collisions at the LHC, the initiallongitudial momentum of the partons is unknown, hence only the missing energy in thetransversal plane, /ET can be used for the WIMP search. The ATLAS [10] and CMS [11]detectors at the LHC were designed to search for the Higgs particle, for new physicsbeyond the SM and as well as for precision tests of the Standard Model. Astrophysicaluncertainties are not present in collider results. However, the very small time a particlespends in the detector offers a huge obstacle in the process of identifying DM particlesfrom collider data alone.

As pair-production of WIMPs of the type qq −→ χχ where χ is the DM particle cannotbe detected by the detectors, the most generic approach to search for WIMP productionat a hadron collider is to search for pair-production associated with initial (or final) stateradiation

qq −→ χχ+X (6.20)

with ′X ′ being a γ, Z or W -boson, or a gluon. Till now, the collider searches using mono-jets or mono-photons accompanied by /ET remained fruitless [262–264]. To allow mean-ingful comparison with direct detection experiment some benchmark scenarios have beenproposed (see e.g, [265–267] and references therein). Assuming that the WIMP is a Diracfermion and that the mediator couples to all quarks with the same strength, it is possibleto compare collider and direct detection searches [173].

The various dark matter search channels presented here are thus highly complemen-tary to one another. While the current generation of collider searches can probe very low

105


WIMP masses, and up to masses of ∼ 1 TeV, direct and indirect detection experimentscan access the WIMP mass region up to 10 TeV and above.

From the above discussions it is evident that various experiments point towards awide range of different DM mass and no definite conclusion has been reached till date.Naturally, a large number of theoretical models have been proposed to solve the DMmystery. We now proceed to propose one such model for dark matter inspired by thetype III seesaw mechanism in the next chapter.

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Chapter 7

A dark matter candidate in an extendedtype III seesaw scenario

7.1 Introduction

We have seen that the Standard Model (SM) of particle physics has been extremely suc-cessful in explaining almost all experimental results. However, extension of the SMis widely expected, to address some yet unexplained observations, which include tinymasses and the mixing pattern of neutrinos and the existence of dark matter (DM) in theUniverse as discussed in the previous chapters.

We have already discussed in Chapter 3 that, one possible explanation of small neu-trino masses can be obtained by adding Y = 0, SU(2)L lepton triplets to the SM. The activeneutrino mixes with the neutral member of such a fermion triplet, allowing ∆L = 2 Majo-rana mass terms for the neutrinos. The heavy Majorana mass of the neutral member of thefermion triplet ensures smallness of the neutrino mass via type-III seesaw mechanism. Asneutrino oscillation experiments provide information only on neutrino mass-squared dif-ferences, the observed data can be explained if at least two neutrinos are massive. Henceneutrino masses in the type-III seesaw mechanism can be explained if we have at leasttwo fermion triplets.

On the other hand, we have seen in the previous chapter that there is strong empiricalevidence for the existence of dark matter in the Universe. A simple and attractive candi-date for DM should be a new elementary particle that is electrically neutral and stable oncosmological scales. In this work, we would like to explore a possibility where the neutralmember of a fermion triplet can turn out as our missing link.

Our proposal is inspired by the type-III seesaw model for neutrino masses. As twofermion triplets are required to generate neutrino masses, we need to introduce a third

107

CHAPTER 7. A DARK MATTER CANDIDATE IN AN EXTENDED TYPE III SEESAWSCENARIO

triplet to take care of the DM candidate. In contrast with the two that generate neutrinomasses, this triplet needs to be protected by a Z2 symmetry to ensure stability of the DMparticle. The SU(2) symmetry will in general demand the charged and neutral compo-nents of the fermion triplet to be degenerate. However, we need a mass splitting betweenthe charged and neutral component to obtain a DM particle of mass of the order of theelectroweak scale. In some models this mass splitting between the charged and neu-tral components of the fermion triplet have been obtained through radiative correction.Refs. [269] and [270] discussed TeV scale DM in the context of fermion triplets using ra-diative correction. The mass of the charged member of the triplet in such a case has to bewell above ∼ TeV, in order to avoid fast t-channel annihilation and a consequent deple-tion in relic density. As a result, such models end up with somewhat inflexible predictionof DM mass in the region 2.5 − 2.7 TeV. Alternative models predicting DM mass in theGeV region open up a new way to look into the DM mystery [271].

We propose, as an alternative, two Z2-odd fermion fields, a Y = 0 triplet and a neutralsinglet that appears as a heavy sterile neutrino. All SM fields and the first two fermiontriplets are even under this Z2 symmetry. Mixing between neutral component of thetriplet and the sterile neutrino can yield a Z2 odd neutral fermion state that is lighter thanall other states in the Z2 odd sector. This fermionic state emerges as the DM candidatein this work [272]. Furthermore, the requisite rate of annihilation is ensured by postulat-ing some Z2 preserving dimension-five operators. These operators, together with the factthat the neutral component of the DM candidate has W couplings, allow a larger regionof the parameter space than what we would have had with a sterile neutrino alone. Thisinterplay brings an enriched DM phenomenology compared to models with only singletor triplets. This, we feel, is a desirable feature of our model given the contradictory claimson the mass of a DM candidate from direct as well as indirect searches. Although choiceof a light sterile neutrino can lead to light DM candidates, in our model, we will workwith DM masses of more than 62.5 GeV to avoid an unacceptably large invisible Higgsdecay width.

We begin by presenting the theoretical framework of our model and introduce themixing term between Z2 odd triplet Σ and sterile neutrino νs in the next section.

7.2 Theoretical Framework

In this section we give a description of our model. We have added three fermion tripletsto the SM Lagrangian. The fermion triplets Σ = (Σ+,Σ0,Σ−) are represented by the 2× 2

108

7.2. THEORETICAL FRAMEWORK

matrix [272]

Σ =

(Σ0/√

2 Σ+

Σ− −Σ0/√

2

), (7.1)

Two of these triplets, which are even under the imposed Z2 symmetry are free to mixwith the usual SM particles and therefore responsible for generation of neutrino massesthrough type III seesaw, explaining the observed mass-squared differences in the neu-trino oscillation experiments. On the other hand, the remaining third triplet does notcontribute to neutrino mass generation through seesaw, because, it is odd under imposedZ2 symmetry. The neutral component of third triplet mix with the Z2 odd sterile neutrinoνs to produce a ’low mass’ dark matter candidate. If νs is light enough and its mixingwith Σ0 is small, the νs-like mass eigenstate can be a viable dark matter candidate.

The added part of Lagrangian with the triplet and the sterile neutrino is given by [121,272–274]: 1

L = Tr[Σai/DΣa]−1

2Tr[ΣaMΣΣc

a+ΣcaM

∗ΣΣa]−Φ†Σb

√2YΣL−L

√2YΣ

†ΣbΦ+i

2νs/∂νs−

1

2Mνsνsνs ,

(7.2)with L ≡ (ν, l)T , Φ ≡ (φ+, φ0)T ≡ (φ+, (v + H + iη)/

√2)T , Φ = iτ2Φ∗, Σc ≡ CΣ

Tand

summation over a and b are implied, with a = 1, 2, 3, b = 1, 2, denote generation indicesfor the triplets. Note that b does not assume the third index as the third generation tripletis odd under Z2.

It should be noted that, in eqn. (7.2), the Yukawa coupling terms for the Z2 odd thirdtriplet and sterile neutrino are prohibited by the exactly conserved Z2 symmetry. Also, inthe covariant derivative of eqn. (7.2), no Bµ terms are present, which in turn restricts theinteraction between fermion triplets and Zµ boson, this is the speciality of the presence ofreal triplets in the theory.

The smallness of the νs – Σ0 mixing can be generated by dimension-five terms. Ingeneral, we add phenomenologically the following terms to the Lagrangian [272]:

L5 =(αΣνsΦ

†ΣΦνs + h.c.)

+ ανsΦ†Φνsνs + αΣΦ†ΣΣΦ (7.3)

where αΣνs , ανs and αΣ are three coupling constants of mass dimension −1.

1We point out that even if the sterile neutrino had a small Majorana mass term, it would not alter ourbroad conclusions in a significant way because of the wide range of mass values we already have consideredfor the sterile neutrino in our study.

109


νs and Σ0 mix to produce two mass eigenstates, χ and Ψ, given by [272]

χ = cos β Σ0 − sin β νs (7.4)

Ψ = sin β Σ0 + cos β νs (7.5)

We denote χ as the lighter mass eigenstate and hence as our candidate for dark matter.The mixing angle β is determined in terms of the five independent parameters present inthe theory, given by [272]

MΣ, Mνs , αΣνs , αΣ and ανs . (7.6)

The Hχχ vertex driving the DM annihilation in the s-channel, is proportional to [272](√2αΣνS cos β sin β + ανS sin2 β +

αΣ

2cos2 β

). (7.7)

In absence of the dimension-five couplings, χ does not couple to the Higgs. HenceHiggs portal DM annihilation can not take place. As mentioned earlier, Z-portal DMannihilation is also not possible. In such a scenario, DM can annihilate via Σ+ exchangein the t-channel to a pair of W bosons. However, in this case, correct relic density can beobtained if the DM mass is larger than 2 TeV as the process is driven by unsuppressedgauge interactions. Introduction of the term αΣΦ†ΣΣΦ induces a mixing between Σ0 andνs which helps us to get a νs-like DM χ. This DM then can self-annihilate via suppressedcouplings with H and W . As we are allowing this dimension-five term, for completenesswe have added the other two dimension-5 terms as well, which in turn give more roomin the parameter space to manoeuvre.

Clearly if all the αs are of the same order, the term αΣΦ†ΣΣΦ will be less effectiveduring annihilation, because for a νs-like dark matter χ, the value of mixing coefficientcos β will be rather small and that will in turn suppress the effect of parameter αΣ on theobservable parameters of the model. However, it cannot be said a priori that the Hχχcoupling as a whole gets suppressed because of the choice of a νs-like DM.

7.3 Results

DM mass depends mainly on the choice of MΣ and Mνs . But it is also dependent on thehigher dimensional parameters. The mass matrix for νs and Σ0 is given by [272]

M =

(Mνs − ανsv2 αΣνsv

2

αΣνsv2 MΣ − αΣv

2

)(7.8)

110

7.3. RESULTS

The DM particle can be identified with the lighter mass eigenstate ofM. In the limit whenMΣ ' Mνs the DM mass is a complicated function of all the five free parameters of ourmodel. In the limit where the mixing between Σ0 and νs is rather small and MΣ Mνs ,the DM mass is approximately expressed as [272],

Mχ ∼Mνs − ανsv2 +(αΣνsv

2)2

MΣ

. (7.9)

The flexibility of our model is that, it allows ‘low’ DM mass for a suitable choice of therelevant parameters, as demonstrated in the following. We use FeynRules 2.0 [275] inconjunction with micrOmegas 3.3.13 [276,277] to compute relic density in our model.

In general, two different situation may arise, both of which contribute to the relic den-sity, they are, (i) MΣ Mνs and (ii) MΣ ' Mνs . We have also noted that, in the secondcase, coannihilation between the charged components of the triplet Σ±, the neutral eigen-state of higher mass Ψ and the dark matter candidate χ becomes important in calculationof relic density.

If the mass difference between the DM and other Z2-odd particles are within 5–10%,it is expected that coannihilation will dominate over DM self-annihilation [219]. Someidea about the various channels of annihilation can be found from the following bench-mark values. When the mass difference between the DM candidate (χ) and other Z2-oddparticles is≈ 200 GeV with the DM mass≈ 875 GeV, then the annihilation channels in de-creasing order of dominance are χχ→ HH , χχ→ ZZ and χχ→ W+W−. Similarly, whenthe mass difference is ≈ 70 GeV and DM mass ≈ 725 GeV, the main annihilation chan-nels in similar order are χχ → W+W−, χχ → HH and χχ → ZZ. For a mass difference≈ 30 GeV and DM mass ≈ 770 GeV, coannihilation takes place via Σ−Ψ→ tb, Σ−Ψ→ ud,Σ−Ψ→ cs, Σ−Σ− → W−W− and ΨΨ→ W+W−. For mass difference≈ 700 GeV (1.3 TeV)and DM mass ≈ 230 GeV (197 GeV), annihilation proceeds along χχ → HH . Finally,for mass difference ≈ 550 GeV with DM mass ≈ 425 GeV, annihilation through both thechannels χχ→ HH and χχ→ W+W− becomes important.

In Fig. 7.1, we fix MΣ = 1 TeV and αΣ = 0.1 TeV−1. We also choose a benchmark valueof Mνs = 200 GeV. We plot DM relic density against the DM mass Mχ. DM mass is variedby changing ανs , but keeping αΣνs fixed at different values. The light shade indicatesthe region that is excluded when Mχ < Mh/2, where Mh is the Higgs mass, due to theconstraint on invisible Higgs decay width. The WMAP [278] and Planck [279] allowedDM relic density being too restrictive, we see from the figure, narrow regions of DM mass∼ 210 GeV and ∼ 240 GeV are allowed. However, for this specific choice of parametersin this plot, only DM mass ∼ 210 GeV is allowed from LUX direct detection constraints.

111


MS=1000 GeV , MΝS=200 GeV

ΑS=0.1 TeV-1

ΑSΝS=2.8 TeV

-1

ΑSΝS=3.0 TeV

-1

ΑSΝS=2.9 TeV

-1

3Σ variation in

Wh2

=0.1198±0.0026

Ex

clu

de

dfr

om

inv

isib

leH

igg

sd

eca

y

MΧ

<M

h2

50 100 150 200 250 3000.00

0.05

0.10

0.15

0.20

M Χ H GeV L

Wh

2

Figure 7.1: Relic density Ωh2 vs. dark matter massMχ keeping αΣνs fixed at 2.8 TeV−1, 2.9 TeV−1

and 3.0 TeV−1. Mχ is varied by changing the remaining parameter ανs . We use MΣ = 1 TeV,αΣ = 0.1 TeV−1, Mνs = 200 GeV. The blue band corresponds to 3σ variation in relic densityaccording to the WMAP + Planck data.

112

7.3. RESULTS

(a) (b)

Figure 7.2: Contour plot in αΣνs – ανs plane for fixed values of MΣ = 1 TeV, Mνs = 200 GeV,αΣ = 0.1 TeV−1. In (a) DM relic density is projected onto the plane, whereas in (b), we projectDM mass (in GeV). In both plots the red solid line represents Ωh2 = 0.1198 and the dashed redlines correspond to the 3σ variation in Ωh2. Darker region corresponds to lower values of Ωh2 orMχ.

In order to find the favoured parameter space which satisfies DM relic density con-straints, in Fig. 7.2(a) we present contour plots of Ωh2 in the αΣνs – ανs plane for MΣ =

1000 GeV and Mνs = 200 GeV. The solid red curve denotes Ωh2 = 0.1198. A 3σ error bandis shown by the dashed red curves. We have varied αΣνs from −6.0 to 6.0 TeV−1 and ανs

from −1.5 to 2.0 TeV−1 to obtain this plot. We have also fixed αΣ = 0.1 TeV−1 for thisplot. In Fig. 7.2(b) we project DM mass Mχ onto the αΣνs – ανs plane for same values ofparameters used in the previous figure. The light green region and the deep green regionin both plots corresponding to Mχ between 205 − 235 GeV indicate the parameter spacewhich satisfy DM relic density constraints as well as WIMP-nucleon cross section boundas imposed by XENON 100 [239] and LUX [244].

In Fig. 7.3(a) we have plotted the WIMP-nucleon cross section vs. dark matter massMχ keeping αΣ fixed at 0.1 TeV−1. Mχ is varied by changing the remaining parameters.For illustration, we have varied MΣ between 300 and 1000 GeV, Mνs between 150 and1000 GeV keeping the mass difference between DM and other Z2 odd fermions greaterthan 200 GeV. The remaining parameters αΣνs and ανs were varied between [−8,+8] TeV−1

and [−0.9,+0.9] TeV−1 respectively. In this case, the DM particle always self-annihilates

113


(a) (b)

Figure 7.3: (a) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ fixed at 0.1TeV−1. Mχ is varied by changing the remaining parameters. In such a scenario, DM annihilatesvia self-annihilation only. Mνs is varied between 150 and 1000 GeV. (b) WIMP-nucleon crosssection vs. dark matter mass Mχ keeping αΣ fixed at 0.1 TeV−1. Mχ is varied by changing theremaining parameters. Here coannihilation between the DM candidate with the charged compo-nents of the triplet Σ±, the neutral eigenstate of higher mass provides the dominating contributionto relic density. In both the figures different direct detection experimental bounds are indicated andthe red points correspond to the 3σ variation in relic density according to the WMAP + Planckdata.

114

7.3. RESULTS

(a) (b)

Figure 7.4: (a) WIMP-nucleon cross section vs. dark matter mass Mχ keeping αΣ fixed at 0.01TeV−1. Mχ is varied by changing the remaining parameters. Here self-annihilation of the DMcandidate provides the dominating contribution to relic density (b) WIMP-nucleon cross sectionvs. dark matter mass Mχ keeping αΣ fixed at 0.01 TeV−1. Mχ is varied by changing the remainingparameters. Here coannihilation between the DM candidate with the charged components of thetriplet Σ±, the neutral eigenstate of higher mass provides the dominating contribution to relicdensity. In both the figures different direct detection experimental bounds are indicated and thered points correspond to the 3σ variation in relic density according to the WMAP + Planck data.

and DM coannihilation does not take place. It is seen that a large region of parameterspace satisfies the bound on WIMP-nucleon cross section as imposed by XENON 100 [239]and LUX [244] experimental data. In Fig. 7.3(b) we have plotted the WIMP-nucleon crosssection vs. dark matter mass Mχ keeping αΣ fixed at 0.1 TeV−1. We varied MΣ between300 and 1000 GeV, Mνs between 270 and 1000 GeV keeping the mass difference betweenDM mass and other Z2 odd fermions always smaller than 30 GeV. The other parametersαΣνs and ανs were varied between [−8,+8] TeV−1 and [−0.9,+0.9] TeV−1 respectively. Inthis case, we noted that coannihilation between the charged components of the triplet Σ±,the neutral eigenstate of higher mass (Ψ) and the dark matter candidate (χ) provides thedominant contributions in the calculation of relic density. Again a large region of param-eter space satisfies the bound on WIMP-nucleon cross section as imposed by XENON 100and LUX experimental data [280] in this case too.

In Fig. 7.4(a), we have plotted the WIMP-nucleon cross section vs. dark matter massMχ keeping αΣ fixed at 0.01 TeV−1. Other parameters αΣνs and ανs were varied between1.2 TeV−1 to 3.0 TeV−1 and −1.1 TeV−1 to 0.03 TeV−1 respectively. In this case, we variedMΣ between 500 and 1500 GeV and Mνs between 150 and 900 GeV. We obtained wide re-gion of DM mass from 128 GeV to 872 GeV that satisfies the correct relic density range and

115


the constrains imposed on WIMP-nucleon cross section, i.e., σSI by XENON 100 and LUXdata. The mass difference between MΣ and Mνs was always kept greater than 150 GeV. Inthis case also the annihilation of DM particle with itself provides the dominating contribu-tion to the calculation of relic density. In Fig. 7.4(b), we have plotted the WIMP-nucleoncross section vs. dark matter mass Mχ keeping αΣ fixed at 0.01 TeV−1. Other parame-ters αΣνs and ανs were varied between −0.08 TeV−1 to 0.94 TeV−1 and −0.009 TeV−1 to0.096 TeV−1 respectively. In this case we varied MΣ between 300 and 1000 GeV and Mνs insuch a way as to make the mass difference between MΣ and Mνs smaller than 55 GeV. Inthis case also coannihilation process between the charged components of the triplet Σ±,the neutral eigenstate of higher mass (Ψ) and the dark matter candidate (χ) provides thedominant contributions in the calculation of relic density. Here also we got a large regionof parameter space that satisfies the correct relic density requirement and the bound im-posed by XENON 100 and LUX data on WIMP-nucleon cross section. In both the plotsthe red points and red regions correspond to those data points that fall within the 3σ

variation in relic density according to the combined WMAP and Planck data.

7.4 Conclusion

To explain smallness of neutrino masses, type-III seesaw mechanism employs heavyfermion triplets. In this chapter, we have tried to investigate a scenario inspired by thismodel to address the dark matter riddle. Previous attempts in this regard employed ra-diative mass splitting of the members of a lepton triplet, yielding DM masses at the TeVscale. We have in addition introduced a sterile neutrino to bring down the DM massscale to ∼ 200 GeV. Both the additional triplet and the sterile neutrino are odd under apostulated Z2 symmetry.

Dark matter direct detection experimental results are rather conflicting. WhileCDMS [227, 281], DAMA [282], CoGeNT [283], CRESST [284] etc. point towards a DMmass of ∼ 10 GeV, XENON 100 and LUX claim to rule out these observations. Given thisscenario, we keep our options open and explore all DM masses allowed by experimentalobservations.

As we are exploring a Higgs portal dark matter model, to respect the invisible Higgsdecay width constraints we have considered DM masses more than Mh/2 ∼ 62.5 GeV. Wehave ensured that our results are consistent with XENON 100 and LUX constraints. Inaddition, we also indicate the constraints on the parameters of our model ensuing fromWMAP and Planck data. In our model, the charged triplet fermions, being Z2 odd, do not

116

7.4. CONCLUSION

couple to SM fermions and as a consequence, can evade detection in the existing collidersincluding LHC.

We have introduced dimension five terms to satisfy the DM relic density constraintswith the help of a sterile neutrino-like DM. The role of these terms is to produce the rightamount of Σ0–νs mixing to get the appropriate DM mass and DM annihilation.

It is possible to satisfy the relic density constraints in a model with a pure sterile neu-trino DM with the dimension five term ανsΦ

†Φνsνs alone, without the triplet fermions.However in this model the allowed parameter space is rather restricted compared to ourmodel which offers a lot of flexibility in DM mass.

In short, we have presented a triplet fermion DM model, which can provide us withDM candidates with masses as low as ≈ 200 GeV. The scenario, however has the flexi-bility to accommodate a higher mass DM candidate as well, and offers a wide region ofparameter space consistent with observations.

117


118

Chapter 8

Summary and conclusions

In the following we make an attempt to discuss in brief the findings of our research in-cluded in this thesis.

The type II seesaw mechanism for neutrino mass generation usually makes use ofone complex scalar triplet. The collider signature of the doubly-charged scalar, the moststriking feature of this scenario, consists mostly in decays into same-sign dileptons orsame-sign W boson pairs. However, certain scenarios of neutrino mass generation, suchas those imposing texture zeros by a symmetry mechanism, require at least two tripletsin order to be consistent with the type II seesaw mechanism.

In Chapter 1, we have briefly described the Standard Model of particle physics and theneed for going beyond it. We presented a partial summary of various neutrino physicsexperiments and their implications on neutrino masses and mixings, some models forneutrino mass generation and two-zero textures of Majorana neutrino mass matrices inChapter 2. We also described how the need for explanation of current neutrino dataserves as a motivation for searching new physics beyond the SM.

In Chapter 3, we described the seesaw models for Majorana mass generation for neu-trinos. In the later part of this chapter we have shown that how the two-zero textures ofneutrino mass matrices become inconsistent with type II seesaw mechanism, when onlyone scalar triplet is present in the scenario.

We developed a model in Chapter 4 with two such complex triplets and showed that,in such a case, mixing between the triplets can cause the heavier doubly-charged scalarmass eigenstate to decay into a singly-charged scalar and a W boson of the same sign.Considering a large number of benchmark points with different orders of magnitude ofthe ∆L = 2 Yukawa couplings, chosen in agreement with the observed neutrino mass andmixing pattern, we demonstrate that H++

1 → H+2 W

+ can have more than 99% branching

119

CHAPTER 8. SUMMARY AND CONCLUSIONS

fraction in the cases where the vacuum expectation values of the triplets are small. It isalso shown that the above decay allows one to differentiate a two-triplet case at the LHC,through the ratios of events in various multi-lepton channels.

In Chapter 5, we reconsidered the scenario mentioned in Chapter 4 with one CP-violating phase present in the scalar potential of the model, and all parameters have beenchosen consistently with the observed neutrino mass and mixing patterns. We foundthat a large phase (& 60) splits the two doubly-charged scalar mass eigenstates widerapart, so that the decay H++

1 → H++2 h is dominant (with h being the 125 GeV scalar).

We identified a set of benchmark points where this decay dominates. This is comple-mentary to the situation, reported in Chapter 4, where the heavier doubly-charged scalardecays as H++

1 → H+2 W

+. We pointed out the rather spectacular signal, ensuing fromH++

1 → H++2 h, in the form of Higgs plus same-sign dilepton peak, which can be observed

at the Large Hadron Collider.We have presented an outline of the Dark Matter problem in Chapter 6. In Chapter 7,

we have considered a model to address the DM riddle inspired by the type III seesawmechanism for neutrino mass generation. The type III seesaw mechanism for neutrinomass generation usually makes use of at least two Y = 0, SU(2)L lepton triplets. Weaugmented such a model with a third triplet and a sterile neutrino, both of which areodd under a conserved Z2 symmetry. With all new physics confined to the Z2-odd sector,whose low energy manifestation is in some higher-dimensional operators, a fermionicdark matter candidate is found to emerge. We identified the region of the parameterspace of the scenario, which is consistent with all constraints from relic density and directsearches, and allows a wide range of masses for the dark matter candidate.

120

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